MMassless Phases for the Villain model in d ≥ (P. Dario) School of Mathematical Sciences, Tel Aviv University Ramat Aviv, Tel Aviv69978, Israel
E-mail address : [email protected] (W. Wu) Statistics department, University of Warwick, Coventry CV4 7AL, UK
E-mail address : [email protected] a r X i v : . [ m a t h - ph ] F e b bstract. We consider the classical Villain rotator model in Z d , d ≥ ∣ x ∣ − d , with an algebraic rate ofconvergence. We also obtain the same asymptotic decay separately for the transversal two-point functions.This quantifies the spontaneous magnetization result for the Villain model at low temperature, and rigorouslyestablishes the Gaussian spin-wave conjecture in dimension d ≥
3. We believe that our method extends tofinite range interactions and to other abelian spin systems and abelian gauge theory in d ≥ ontents Chapter 1. Introduction 51. General overview and statement of the main result 52. Strategy of the proof 73. First order expansion of the two point-functions 104. Open questions 105. Organization of the paper 11Chapter 2. Preliminaries 131. Notations and assumptions 132. Discrete differential forms 183. Convention for constants and exponents 22Chapter 3. Duality and Helffer-Sj¨ostrand representation 231. From Villain model to solid on solid model 232. Brascamp-Lieb inequality 283. Coupling the finite volume Gibbs measures 304. The Helffer-Sj¨ostrand PDE 37Chapter 4. First-order expansion of the two-point function 451. Preliminary notations 462. Removing the terms X sin cos , X cos cos and X sin sin P f X sin cos , X cos cos and X sin sin
5. Treating the error term E q ,q Introduction
1. General overview and statement of the main result
In this paper, we obtain precise low temperature properties of certain classical rotator models in theEuclidean lattice Z d , d ≥
3. A canonical model of rotator is the XY model, which assigns to each x ∈ Z d a spintaking value in S with the corresponding angle θ ( x ) ∈ (− π, π ] . The XY model is defined formally as a Gibbsmeasure with Hamiltonian given by H XY ∶= − ∑ x ∼ y cos ( θ ( x ) − θ ( y )) . The classical Villain model is another canonical two-component spin model which is closely related to theXY model [ ]. Given a finite cube ◻ ⊆ Z d , we denote by ◻ ○ its interior, ∂ ◻ its boundary and E (◻) its edgeset. The Villain model on the cube ◻ with zero boundary condition is given by the following Gibbs measure(1.1) dµ Vβ, ◻ , ( θ ) ∶= Z − ◻ , ∏ ( x,y )⊆ E (◻) v β ( θ ( x ) − θ ( y )) ∏ x ∈ ∂ ◻ δ ( θ ( x )) ∏ x ∈◻ ○ [− π,π ) ( θ ( x )) dθ where v β ( θ ) = ∑ m ∈ Z exp (− β ( θ + πm ) ) is the heat kernel on S and Z ◻ , is the normalization constant that makes (1.1) a probability measure. Theexpectation with respect to the Gibbs measure (1.1) is denoted by ⟨⋅⟩ µ Vβ, ◻ , . We also define the spin variable,which takes value on the unit circle, by S x = ( cos θ ( x ) , sin θ ( x )) . By the θ → − θ symmetry, we have ⟨ S ⋅ S x ⟩ µ Vβ, ◻ , = ⟨ e i ( θ ( )− θ ( x )) ⟩ µ Vβ, ◻ , . It is known that, as a consequence of correlation inequalities [
44, 18, 61 ], the thermodynamic limit of themeasures (1.1) exists as ◻ → Z d . We denote by µ Vβ the corresponding infinite volume Gibbs measure. It isclear that the quantity ⟨ S ⋅ S x ⟩ µ Vβ = ⟨ e i ( θ ( )− θ ( x )) ⟩ µ Vβ is invariant under the rotations θ → θ + α mod 2 π .There has been long standing interests in the low temperature behavior of the XY and the Villain model.A simple heuristics suggests that, as the temperature goes to zero, the spins tend to align with each other so asto minimize the Hamiltonian. Since exp ( β cos ( δθ )) ≈ ∑ m ∈ Z exp (− β ( δθ + πm ) ) and cos 2 π ( δθ ) ≈ − ( δθ ) / ] (see also [ ]) and was referred to as the Gaussianspin-wave approximation. By further making connections between the rotational symmetry of these modelsand the recurrence/transience property of simple random walks, it was proved in [ ] that for d ≤
2, there isno spontaneous magnetization at any strictly positive temperature. A related argument was applied in [ ] toshow that for d ≥
3, with appropriate boundary conditions, the SO ( ) symmetry is broken at low temperature,i.e., there is spontaneous magnetization ⟨ cos θ ⟩ µ Vβ >
0, but there is no spontaneous magnetization at hightemperature (and thus there is a phase transition).Quantitative improvements of the spontaneous magnetization result were established in the 1980s. If webelieve that the low temperature behavior of the rotator models is like the one of a Gaussian free field, then aGaussian computation yields that for d ≥ ⟨ e i ( θ ( )− θ ( x )) ⟩ µ Vβ ≈ exp ( δ − δ x , − β ∆ − ( δ − δ x )) = C ( β ) + C ( β )∣ x ∣ d − + o ( ∣ x ∣ d − ) . For d =
2, however, a Gaussian computation yields that for large β ,(1.3) ⟨ e i ( θ ( )− θ ( x )) ⟩ µ Vβ ≈ C ∣ x ∣ − πβ + o (∣ x ∣ − πβ ) . The algebraic decay of correlation indicates the so-called Kosterlitz-Thouless transition in two dimensions [ ].In d =
2, the algebraic decay of correlations is expected to be valid for all temperatures below the criticaltemperature corresponding to the K-T transition, and above which one expects exponential decay of correlations.The Gaussian spin-wave approximation (1.3) for d = β sufficiently large, rigorous(but non-optimal) power law upper and lower bounds for the two-point function were established in thecelebrated works [ ] and [ ].For d ≥
3, Fr¨ohlich and Spencer [ ] observed that the classical Villain model in Z d can be mapped, viaduality, to a statistical mechanical model of lattice Coulomb gas, with local neutrality constraints. Theyfurther employed a one step renormalization argument and gave the following next order description of thetwo-point function at low temperature. Proposition ]) . Let µ Vβ be the thermodynamic limit of the Villain model in Z d , for d ≥ . Thereexist constants β = β ( d ) , c = c ( β, d ) , such that for all β > β , ⟨ S ⋅ S x ⟩ µ Vβ = c + O ( ∣ x ∣ d − ) Moreover, as β → ∞ , exp ( δ − δ x , − β ∆ − ( δ − δ x )) ≥ ⟨ S ⋅ S x ⟩ µ Vβ ≥ exp ( δ − δ x , (− β + o ( β )) ∆ − ( δ − δ x )) . This suggests that the truncated two-point function may be related to a massless free field in R d , whichcorresponds to the emergence of a (conjectured) Goldstone Boson. Similar results were also obtained for theAbelian gauge theory in d =
37, 51 ]). Kennedy and King in [ ] obtained a similar low temperatureexpansion for the Abelian Higgs model, which couples an XY model with a gauge fixing potential. Theirproofs rely on a different approach, via a transformation introduced by [ ] and a polymer expansion.It is also of much interest to justify the spin-wave conjecture separately for the longitudinal and transversaltwo-point functions of the rotator models. The best known result is due to Bricmont et. al. [ ], where theyperform a low temperature expansion of the truncated two-point function of the XY model and obtain thatthere exist c > c >
0, such that for sufficiently large β , c β ∣ x ∣ d − ≤ ⟨ sin θ ( ) sin θ ( x )⟩ µ XYβ ≤ c β ∣ x ∣ d − . The proof of [ ] relies on a combination of the infrared bound, a Mermin-Wagner type argument, andcorrelation inequalities, and is therefore restricted to the cubic lattice with nearest neighbor interactions.The main result of our paper, stated below, resolves the long standing spin-wave conjecture for the Villainmodel in d ≥
3, by obtaining the precise asymptotics of the two-point functions at low temperature.
Theorem . Let µ Vβ be the thermodynamic limit of the Villain model in Z d , for d ≥ . There existconstants β = β ( d ) , c = c ( β, d ) , c = c ( β, d ) , c = c ( β, d ) , and α = α ( d ) > such that for all β > β , (1.4) ⟨ sin θ ( ) sin θ ( x )⟩ µ Vβ = c ∣ x ∣ d − + O ( ∣ x ∣ d − + α ) , and (1.5) ⟨ S ⋅ S x ⟩ µ Vβ = c + c ∣ x ∣ d − + O ( ∣ x ∣ d − + α ) . It will be clear from the proof of Theorem 1 (see the presentation in Chapters 3 and 4), that the constant c is a small correction to the prediction C from the free field computation (1.2) (and satisfies ∣ c − C ∣ ≤ e − cβ / ).Indeed, as observed in [ ], the emergent massless free field that leads to the ∣ x ∣ − d term is contributedfrom a free field (1.2), plus a vortex correction (see Chapter 3, in particular (1.5) there). Our result givesthe covariance structure of the emergent massless free field. In fact, we see that c = ⟨ cos θ ⟩ µ Vβ . We maycharacterize c from the homogenization of a family of elliptic PDEs (see Chapter 4), and thus identify thecorrelation structure of the continuum free field. . STRATEGY OF THE PROOF 7 The proof in this paper is robust and can be adapted to general non-cubic lattices and general finite rangeinteractions. It also applies to boundary conditions other than Dirichlet. For example, we may consider theGibbs states arising as the infinite volume limit of the Villain models with Neumann or periodic boundaryconditions. The same proof applies (except for several changes in the boundary conditions in Chapter 3 whenwe apply the duality) and we can also prove Theorem 1 for these Gibbs states. With the same approach,but more elaborate estimates, it is possible to further generalize our result and prove that the spin fieldcos θ x − ⟨ cos θ x ⟩ µ Vβ converges in distribution to a Gaussian free field, with the covariance structure given in (1.5).Finally, the method extends to more general models such as the higher dimensional abelian spin models andabelian lattice gauge theorems, and these may lead to several subsequent works.
2. Strategy of the proof2.1. Sine-Gordon representation and polymer expansion.
The spin wave computation (1.2) isonly heuristic and does not give the correct constants C , C . The main problem for the spin wave is thatit ignores the formation of vortices, that are defined on the faces of Z d . Kosterlitz and Thouless [ ] gave aheuristic argument, that indicates that the vortices are interacting like a neutral Coulomb gas, taking integervalued charges.Our proof of Theorem 1 starts from an insight of Fr¨ohlich and Spencer [ ], which makes this observationrigorous. In particular, the two-point function of the Villain model in Z d , d ≥ ( Z d ) . Byperforming a Fourier transform with respect to the charge variable, a classical random field representation ofthe Coulomb gas, known as the sine-Gordon representation is obtained. When the temperature is low enough,the charges are essentially supported on short range dipoles, therefore a one-step renormalization argumentand a cluster expansion can be applied, following the presentation of [ ], in order to reduce the effectiveactivity of the charges. This leads to an effective, vector-valued random interface model with infinite range anduniformly convex potential. The question of the asymptotic behavior of the two-point function is thus reducedto the question of the quantitative understanding of the large-scale properties of the random interface model. The study of the large-scale properties of the randominterface model starts from the insight of Naddaf and Spencer [ ] that the fluctuations of the field are closelyrelated to an elliptic homogenization problem for the Helffer-Sj¨ostrand equation [
53, 70 ]. This approachhas been used by Giacomin, Olla and Spohn in [ ] to prove that the large-scale space-time fluctuationsof the field is described by an infinite dimensional Ornstein-Uhlenbeck process and by Deuschel, Giacominand Ioffe to establish concentration properties and large deviation principles on the random surface (seealso Sheffield [ ], Biskup and Spohn [ ], Cotar and Deuschel [ ] and Adams, Buchholz, Koteck`y andM¨uller [ ] for an extension of these results to some non-convex potentials). An important component ofthe strategy presented in many of the aforementioned articles relies on a probabilistic approach: one can,through the Helffer-Sj¨ostrand representation, reduce the problem to a question of random walk in dynamicrandom environment, and then prove properties on this object, e.g. invariance principles, using the resultsKipnis and Varadhan [ ], or annealed upper bounds on the heat kernel (see Delmotte and Deuschel [ ]).However, the results obtained so far using this probabilistic approach are not quantitative, and it cannot beeasily implemented to study nonlinear functionals of the field. A more analytical approach was developedby Armstrong and Wu in [ ], where they extend and quantify the homogenization argument of Naddaf andSpencer [ ], resolved an open question posed by Funaki and Spohn [ ] regarding the C regularity of surfacetension, and also positively resolve the fluctuation-dissipation conjecture of [ ].The approach followed in this article is the analytical one, and fits into the program developed in [
65, 6 ]on homogenization for the random interface models. Since the sine-Gordon representation and the polymerexpansion give a random interface model valued in the vector space R ( d ) (which corresponds to the dimensionof the space of discrete 2-forms on Z d used to derive the dual Villain model as explained in Section 2.1 andChapter 3) with long range and uniformly convex potential, an application of the strategy of Naddaf andSpencer [ ] to this model gives a Helffer-Sj¨ostrand operator of the form(2.1) ∆ φ + L , which is an infinite dimensional elliptic operator acting on functions defined in the space Ω × Z d (see (4.1) inChapter 3 for the precise definition of this operator), where Ω is the space of functions from Z d to R ( d ) inwhich the vector-valued random interface considered in this article is valued. The operator ∆ φ is the (infinite
1. INTRODUCTION dimensional) Laplacian computing derivatives with respect to the height of the random surface and L isan operator associated to a uniformly elliptic system of equations with infinite range (and with exponentialdecay on the size of the long range coefficients) on the discrete lattice Z d . The analysis of these systemsrequires to overcome some difficulties; a number of properties which are valid for elliptic equations, and usedto study the random interface models, are known to be false for elliptic systems. It is for instance the casefor the maximum principle, which implies there is no the random walk representation for this model, the DeGiorgi-Nash-Moser regularity theory for uniformly elliptic and parabolic PDE (see [
66, 28 ], [ , Chapter 8]and the counterexample of De Giorgi [ ]) and the Nash-Aronson estimate on the heat kernel (see [ ]).To resolve this lack of regularity, we rely on a perturbative argument and make use of ideas from Schaudertheory (see [ , Chapter 3]); we leverage on the fact that the inverse temperature β is chosen very large sothat the elliptic operator L can be written L ∶= − β ∆ + L pert , where the operator L pert is a perturbative term; its typical size is of order β − ≪ β − . One can thus provethat any solution u of the equation (2.1) is well-approximated on every scale by a solution u of the equation∆ φ − β ∆ for which the regularity can be easily established. It is then possible to borrow the strong regularityproperties of the function u and transfer it to the solution of (2.1). This strategy is implemented in Chapter 5and allows us to prove the C , − ε -regularity of the solution of the Helffer-Sj¨ostrand equation, and to deducefrom this regularity property various estimates on other quantities of interest (e.g, decay estimates on theheat kernel in dynamic random environment, decay and regularity for the Green’s function associated to theHelffer-Sj¨ostrand operator). The regularity exponent ε depends on the dimension d and the inverse temperature β , and tends to 0 as β tends to infinity; in the perturbative regime, the result turns out to be much strongerthan the C ,α -regularity provided by the De Giorgi-Nash-Moser theory (for some tiny exponent α >
0) in thecase of elliptic equations, and implies sufficient mixing of the solution to the Helffer-Sj¨ostrand equation.
The main difficulty in the establishment of Theorem 1 is that sincethe Villain model is not exactly solvable, the dependence of the constants c and c on the dimension d and theinverse temperature β is highly non explicit; one does not expect to have a simple formula for these coefficientsbut it is necessary to analyze them in order to prove the expansions (1.4) and (1.5). This is achieved by usingtools from the theory of quantitative stochastic homogenization.This theory is typically interested in the understanding of the large-scale behavior of the solutions of theelliptic equation(2.2) − ∇ ⋅ a ( x )∇ u = , where a is a random, uniformly elliptic coefficient field that is stationary and ergodic. The general objective isto prove that, on large scales, the solutions of (2.2) behave like the solutions of the elliptic equation(2.3) − ∇ ⋅ a ∇ u = , where a is a constant uniformly elliptic coefficient called the homogenized matrix . The theory was initiallydeveloped in the 80’s, in the works of Kozlov [ ], Papanicolaou and Varadhan [ ] and Yurinski˘ı [ ]. DalMaso and Modica [
22, 23 ] extended these results a few years later to nonlinear equations using variationalarguments inspired by Γ-convergence. All of these results rely on the use of the ergodic theorem and aretherefore purely qualitative.The main difficulty in the establishment of a quantitative theory is the question of the transfer of thequantitative ergodicity encoded in the coefficient field a to the solutions of the equation. This problem wasaddressed in a satisfactory fashion for the first time by Gloria and Otto in [
47, 48 ], where, building upon theideas of [ ], they used spectral gap inequalities (or concentration inequalities) to transfer the quantitativeergodicity of the coefficient field to the solutions of (2.2). These results were then further developed in [
49, 50 ]and also in collaboration with Neukamm in [ ] and [ ].Another approach, which is the one pursued in this article, was initiated by Armstrong and Smart in [ ],who extended the techniques of Avellaneda and Lin [
9, 10 ], the ones of Dal Maso and Modica [
22, 23 ] andobtained an algebraic, suboptimal rate of convergence for the homogenization error of the Dirichlet problemassociated to the nonlinear version of the equation (2.2). These results were then improved in [
3, 4, 5 ] to obtainoptimal rates. Their approach relies on mixing conditions on the coefficient fields and on the quantification ofthe subadditivity defect of dual convex quantities (see Chapter 6). The paper [ ] also addressed for the first . STRATEGY OF THE PROOF 9 time a large-scale regularity theory for the solutions of (2.2), which was later improved in the works of [ ] and[ ]. An extension of the techniques of [ ] to the setting of differential forms (which also appear in this articlein the dual Villain model) can be found in [ ], and to the uniformly convex gradient field model in [ ].To prove Theorem 1, we apply the techniques of [ ] to the Helffer-Sj¨ostrand equation to prove thequantitative homogenization of the mixed derivative of the Green’s function associated to this operator. Thestrategy can be decomposed into two steps.The first one relies on the variational structure of the Helffer-Sj¨ostrand operator and is the main subjectof Chapter 6: following the arguments of [ , Chapter 2], we define two subadditive quantities, denoted by ν and ν ∗ . The first one corresponds to the energy of the Dirichlet problem associated to the Hellfer-Sj¨ostrandoperator (2.1) in a domain U ⊆ Z d and subject to affine boundary condition, the second one corresponds to theenergy of the Neumann problem of the same operator with an affine flux. Each of these two quantities dependson two parameters: the domain of integration U and the slope of the affine boundary condition, denoted by p (for ν ) and p ∗ (for ν ∗ ). These energies are quadratic, uniformly convex with respect to the variables p and p ∗ ,and satisfy a subadditivity property with respect to the domain U ; in particular, an application of Fekete’sLemma shows that these quantities converge as the size of the domain tends to infinity ν ( U, p ) —→ ∣ U ∣→∞ p ⋅ a p and ν ∗ ( U, p ∗ ) —→ ∣ U ∣→∞ p ∗ ⋅ a ∗ p ∗ . The coefficient a obtained this way plays a similar role as the homogenized matrix in (2.3); in the case of thepresent random interface model, it gives the coefficient obtained in the continuous (homogenized) Gaussianfree field which describes the large-scale behavior of the random surface as established by Naddaf and Spencerin [ ]. The objective of the proofs of Chapter 6 is to quantify this convergence and to obtain an algebraicrate: we show that there exists an exponent α > ◻ ⊆ Z d of size R > ∣ ν ( U, p ) − p ⋅ a p ∣ + ∣ ν ( U, p ∗ ) − p ∗ ⋅ a ∗ p ∗ ∣ ≤ CR − α . The strategy to prove the quantitative rate (2.4) is to use that the maps p ↦ ν ( U, p ) and p ∗ ↦ ν ∗ ( U, p ∗ ) are approximately convex dual. We use a multiscale argument to prove that, by passing from one scale toanother, the convex duality defect must contract (in particular, it is equal to 0 in the infinite volume limit, i.e., a ∗ = a − ). More precisely we show that the convex duality defect can be controlled by the subadditivity defect,and then iterate the result over all the scales from 1 to R to obtain (2.4). As a byproduct of the proof, weobtain a quantitative control on the sublinearity of the finite-volume corrector defined as the solution of theDirichlet problem: given an affine function l p of slope p and a cube ◻ ⊆ Z d of size R , { ( ∆ φ + L) ( l p + χ ◻ ,p ) = × ◻ ,χ ◻ ,p = × ∂ ◻ . This estimate takes the following form(2.5) 1 R ∥ χ ◻ ,p ∥ L (◻ ,µ β ) ≤ CR α , where the average L -norm is considered over both the spatial variable and the random field.We note that, contrary to the case of the homogenization of the elliptic equation (2.2), the subadditivequantities are deterministic objects and are applied to the operator (2.1) which is essentially infinite dimensional.While the proofs of [ , Chapter 2] rely on a finite range dependence assumption to quantify the ergodicity ofthe coefficient field, we rely here on the regularity properties of the Helffer-Sj¨ostrand operator to prove sufficientdecorrelation estimates on the field to obtain the algebraic rate of convergence stated in (2.4). The sameissues were addressed in the work of Armstrong and Wu [ ], to study the ∇ φ -model and prove C -regularityof the surface tension conjectured by Funaki and Spohn [ ]; the arguments presented there are somewhatsimilar to ours but with a distinct difference: they rely on couplings based on the probabilistic interpretationof the equation to obtain sufficient decorrelation of the gradient field. In the present paper, we rely on theobservation that the differential with respect to the field ∂u , where u is a solution to the Helffer-Sj¨ostrandequation, solves a differentiated Helffer-Sj¨ostrand equation introduced in Section 4 of Chapter 5, and thedecorrelation follows from the regularity theory for the differentiated Helffer-Sj¨ostrand operator.The second step in the argument, which extends the results of [ ], is to prove quantitative homogenizationof the mixed derivative of the Green’s function associated to the Helffer-Sj¨ostrand operator (2.1). In thesetting of the divergence from elliptic operator (2.2), the properties of the Green’s function are well-understood; moment bounds on the Green’s function, its gradient and mixed derivative are proved in [ ], [ ] and [ ].Quantitative homogenization estimates are proved in [ , Chapters 8 and 9] and in [ ]. The argument relieson a common strategy in stochastic homogenization: the two-scale expansion. It is implemented as follows:given a function f ∶ Ω → R which depends on the field and satisfies some suitable regularity assumptions (e.g. f depends on finitely many height variables of the random surface, is smooth and compactly supported), thelarge-scale behavior of the fundamental solution G f ∶ Ω × Z d → R ( d )×( d ) of the ( d ) -dimensional elliptic system ( ∆ φ + L) G f = f δ , is described by the (deterministic) fundamental solution G ∶ Z d → R ( d )×( d ) of the homogenized elliptic system −∇ ⋅ a ∇ G = ⟨ f ⟩ µ β δ , where the notation µ β is used to denote the law of the random surface and ⟨ f ⟩ µ β denotes the expectationof the function f with respect to the probability measure µ β . The proof of this result relies on a two-scaleexpansion for systems of equations: we select a suitable cube ◻ ⊆ R d and define the function H ⋅ k ∶= G ⋅ k + d ∑ i = ( d ) ∑ j = χ ◻ ,e ij ∇ i G jk . We then compute the value of ( ∆ φ + L) H and prove, by using the quantitative information obtained on thecorrector (2.5), that this value is small in a suitable functional space. This argument shows that the function H (resp. its gradient) is quantitatively close to the functions G f (resp. its gradient). Once this is achieved,we can iterate the argument to obtain a quantitative homogenization result for the mixed derivative of theGreen’s function following the description given at the beginning of Chapter 6. The overall strategy is similarto the one in the case of the divergence form elliptic equations (2.2) but a number of technicalities need to betreated along the way: ● The infinite dimensional Laplacian ∆ φ needs to be taken into account in the analysis; ● The elliptic operator L given in the model has infinite range; ● One needs to homogenize an elliptic system instead of an elliptic PDE.While the first point has been successfully addressed by [ ], the last two points are intrinsic to the Coulombgas representation of the dual Villain model. Overcoming these difficulties requires new adaptation of themethods developed in [ ] and [ ].
3. First order expansion of the two point-functions
The first order expansion of the two-point function is obtained by post-processing all the argumentsabove. We first use the sine-Gordon representation and the polymer expansion to reduce the problem to theunderstanding of the large scale behavior of a vector-valued random surface model, whose Hamiltonian is asmall perturbation of the one of a Gaussian free field. We then use the ideas of Naddaf and Spencer [ ]and Schauder regularity theory (through a perturbative argument) to obtain a precise understanding of thecorrelation structure of the random field. Unfortunately, the regularity obtained this way is not strong enoughto establish Theorem 1 and to obtain a sharp control on the large-scale behavior of the solutions of theHelffer-Sj¨ostrand equation, we adapt the recent techniques in quantitative stochastic homogenization developedin [ ]. We can then combine this result with the regularity theory and the rotation and symmetry invarianceof the Villain model to prove Theorem 1. The proof of this result requires to analyze a number of terms toisolate the leading order terms and to estimate quantitatively the lower order ones. It is rather technical andis split into two chapters: in Chapter 4 we present a detailed sketch of the argument, isolate the leading orderfrom the lower order terms and state the estimates on each of these terms; Chapter 8 is devoted to the proof ofthe technical estimates and the estimates of the various terms obtained in Chapter 4 using the results provedin Chapters 5, 6 and 7.
4. Open questions
Finally, we discuss some open questions regarding the low temperature phase of the classical XY model in Z d , d ≥
3. The Gaussian spin-wave approximation predicts that the two-point function of the XY model alsoadmits a low temperature expansion stated in Theorem 1. It is believable that our method can be adaptedto resolve this conjecture for the XY model. The main challenge is that, unlike the Villain model, when . ORGANIZATION OF THE PAPER 11 passing to the dual model the two-point function cannot be factorized as a Gaussian contribution and a vortexcontribution (see Chapter 3, (1.5)), thus requires a new idea for renormalization.
5. Organization of the paper
This paper is organized as follows. In the next chapter we introduce some preliminary notations. InChapter 3, we recall the dual formulation of the Villain model in terms of a vector-valued random interfacemodel, based on the ideas of Fr¨ohlich and Spencer [ ] and following the presentation of Bauerschmidt [ ].We also establish the existence of a thermodynamic limit for the dual Villain model by coupling the Langevindynamics of the finite volume models following the technique of Funaki and Spohn [ ]. We then derive theHelffer-Sj¨ostrand equation for the dual model and state the main regularity estimates on the Green’s functionproved in Chapter 5 and the quantitative homogenization of the mixed derivative of the Green’s functionproved in Chapters 6 and 7. In Chapter 4, we sketch the proof of the main theorem, assuming the C , − ε regularity for the solutions of the Helffer-Sj¨ostrand equation (established in Chapter 5), and the quantitativehomogenization of the mixed derivative of the Green’s function (established in Chapter 7). In Chapter 6, weintroduce the subadditive energy quantities and show by a multiscale iterative argument that they converge atan algebraic rate. Finally in Chapter 8, we give detailed proofs of the claims in Chapter 4. Acknowledgments.
P.D. is supported by the Israel Science Foundation grants 861/15 and 1971/19 andby the European Research Council starting grant 678520 (LocalOrder). W.W. is supported in part by theEPSRC grant EP/T00472X/1. We thank T. Spencer for many insightful discussions that inspired the project,R. Bauerschmidt for kindly explaining the arguments in [ ], and S. Armstrong for many helpful discussions.We also thank S. Armstrong and J.-C. Mourrat for helpful feedbacks on a previous version of the paper.HAPTER 2 Preliminaries
1. Notations and assumptions1.1. General notations and assumptions.
We work on the Euclidean lattice Z d in dimension d ≥ x, y ∈ Z d are neighbors, and denote it by x ∼ y , if ∣ x − y ∣ =
1. We denote by e , . . . , e k the canonical basis of R d . We denote by ∣⋅∣ the standard Euclidean norm on the lattice Z d . For each integer k ∈ { , . . . , d } , a k -cell of the lattice Z d is a set of the form, for a subset { i , . . . , i k } ⊆ { , . . . , d } and a point x ∈ Z d , { x + k ∑ l = λ l e i l ∈ R d ∶ ≤ λ , . . . , λ k ≤ } . We equip the set of k -cells with an orientation induced by the canonical orientation of the lattice Z d and denoteby Λ k ( Z d ) the set of oriented k -cells of the lattice Z d . Given a k -cell c k , we denote by ∂c k the boundary ofthe cell; it can be decomposed into a disjoint union of ( k − ) -cells. The values k = , , Z d . We will denote these spaces by V ( Z d ) , E ( Z d ) and F ( Z d ) respectively.Given a subset U ⊆ Z d we define its interior U ○ and its boundary ∂U by the formulas U ○ ∶= { x ∈ U ∶ x ∼ y (cid:212)⇒ y ∈ U } and ∂U ∶= U ∖ U ○ . If the subset U ⊆ Z d is finite, we denote by ∣ U ∣ its cardinality and refer to this quantity as the volume of U .We denote by diam U the diameter of U defined by the formula diam U ∶= sup x,y ∈ U ∣ x − y ∣ . Given a point x ∈ Z d and a radius r >
0, we denote by B ( x, r ) the discrete euclidean ball of center x and radius r . We frequentlyuse the notation B r to mean B ( , r ) .A discrete cube ◻ of Z d is a subset of the form ◻ ∶= x + [− N, N ] d ∩ Z d with x ∈ Z d and N ∈ N . We refer to the point x as the center of the cube ◻ and to the integer 2 N + r >
0, we use the nonstandard convention of denoting by r ◻ the cube r ◻ ∶= x + [− rN, rN ] d ∩ Z d . We denote by Λ k (◻) the set of oriented k -cells qhich are included in the cube ◻ . In the specific cases k = , V (◻) , E (◻) and F (◻) the set of vertices, edges and faces of the cube ◻ respectively.For each integer i ∈ { , . . . , d } ,we denote by h i the reflection of the lattice Z d with respect to the hyperplane { z ∈ Z d ∶ z i = } , i.e., h i ∶= ⎧⎪⎪⎨⎪⎪⎩ Z d → Z d ( z , . . . , z d ) ↦ ( z , . . . , − z i , . . . , z d ) . For each pair of integers i, j ∈ { , . . . , d } with i < j , we denote by h ij the map h i ∶= ⎧⎪⎪⎨⎪⎪⎩ Z d → Z d ( z , . . . , z d ) ↦ ( z , . . . , z j , . . . , z i , . . . , z d ) . We define H the group of lattice preserving transformation to be the group of linear maps generated by thecollections of functions ( h i ) ≤ i ≤ d and ( h ij ) ≤ i < j ≤ d with respect to the composition law.Given three real numbers X, Y ∈ R and κ ∈ [ , ∞) , we write X = Y + O ( κ ) if and only if ∣ X − Y ∣ ≤ κ.
134 2. PRELIMINARIES
For each integer k ∈ N , we let F ( Z d , R k ) be the set offunctions defined on Z d and taking values in R k . Given a function g ∈ F ( Z d , R k ) , we denote by g , . . . , g k itscomponents on the canonical basis of R k and write g = ( g , . . . , g k ) . We define the support of the function g tobe the set supp g ∶= { x ∈ Z d ∶ g ( x ) ≠ } . The oscillation of a function g over a set U ⊆ Z d is defined by the formulaosc U g ∶= sup U g − inf U g. For each exponent α >
0, we define the C ,α -H¨older seminorm of the function g over the set U by ∥ g ∥ C ,α ( U ) ∶= sup x,y ∈ U,x ≠ y ∣ g ( x ) − g ( y )∣∣ x − y ∣ α . For each integer i ∈ { , . . . , d } , we define its discrete i -th derivative ∇ i g ∶ Z d → R k by the formula, for each x ∈ Z d , ∇ i g ( x ) ∶= g ( x + e i ) − g ( x ) , and its gradient ∇ g ∶ Z d → R d × k by the formula(1.1) ∇ g ( x ) = (∇ i g ( x )) ≤ i ≤ d = (∇ i g j ( x )) ≤ i ≤ d, ≤ j ≤ k . We denote by ∇ ∗ i the adjoint gradient defined by the formula ∇ ∗ i g ( x ) = g ( x − e i ) − g ( x ) . A property of thediscrete setting is that the L ∞ -norm of the gradient of a function g is bounded from above by the C ,α -H¨olderseminorm of this function: we have, for any U ⊆ Z d and any exponent α > x ∈ U ∣∇ g ( x )∣ ≤ ∥ g ∥ C ,α ( U ) . The Laplacian ∆ g ∶ Z d → R k is defined by the formula, for each point x ∈ Z d ,(1.2) ∆ g ( x ) = ∑ y ∼ x ( g ( y ) − g ( x )) . For each integer n ∈ N , one can consider the iteration ∇ n on the gradient and ∆ n of the Laplacian. Wenote that these discrete operators have range n and 2 n respectively, i.e., given a point x ∈ Z d and a function u ∶ Z d → R k one can compute the value of ∇ n u ( x ) (resp. ∆ n u ( x ) ) by knowing only the values of u inside theball B ( x, n ) (resp. B ( x, n ) ). For each function g ∶ Z d × Z d → R k , we denote by ∇ x and ∇ y the gradients withrespect to the first and second variable respectively, i.e., for each point ( x, y ) ∈ Z d × Z d , we write ∇ x g ( x, y ) = ( g j ( x + e i , y ) − g j ( x, y )) ≤ i ≤ d, ≤ j ≤ k and ∇ y g ( x, y ) = ( g j ( x, y + e i ) − g j ( x, y )) ≤ i ≤ d, ≤ j ≤ k . We similarly define the i -th derivatives ∇ i,x and ∇ i,y and the Laplacians ∆ x and ∆ y with respect to the firstand second variables.Given two functions f, g ∶ Z d → R k and a point x ∈ Z d , we define the scalar product f ( x ) ⋅ g ( x ) ∶=∑ di = f k ( x ) g k ( x ) . To ease the notation, we may write f ( x ) g ( x ) to mean f ( x ) ⋅ g ( x ) . We define the L -scalarproduct (⋅ , ⋅) according to the formula(1.3) ( f, g ) = ∑ x ∈ Z d f ( x ) g ( x ) , We restrict this scalar product to a set U ⊆ Z d and define, for any pair of functions f, g ∶ U → R k ,(1.4) ( f, g ) U ∶= ∑ x ∈ U f ( x ) g ( x ) . We define the divergence operator ∇⋅ on vector valued functions: given a function F ∶ Z d → R d × k , we denoteby ∇ ⋅ F ∶ Z d → R k the unique function which satisfies, for each compactly supported function g ∶ Z d → R k , ( F, ∇ g ) = − (∇ ⋅ F, g ) . If we use the definition (1.1) and denote by ( F ij ) ≤ i ≤ d, ≤ j ≤ k the components of the function F , then we havethe identity ∇ ⋅ F = (∑ di = ∇ ∗ i F ij ) ≤ j ≤ k .Given an integer l ∈ N and a function h ∶ Z d → R l , we denote by g ⊗ h ∶ Z d → R k × l the tensor productbetween the two functions h and g ; it is defined by the formula, for each point x ∈ Z d ,(1.5) g ⊗ h ( x ) ∶= ( g i ( x ) h j ( x )) ≤ i ≤ k, ≤ j ≤ l . . NOTATIONS AND ASSUMPTIONS 15 This notation allows to expand gradients of products of functions: for each function u ∶ Z d → R , one has(1.6) ∇( ug )( x ) = ∇ u ⊗ g ( x ) + u ( x )∇ g ( x ) . Given a bounded subset U ⊆ Z d , we define the average of g over the set U by the formula ( g ) U ∶= ∣ U ∣ ∑ x ∈ U g ( x ) ∈ R k . For each real number p ∈ [ , ∞) and each subset U ⊆ Z d , we define the L p ( U ) -norm ∥ g ∥ L p ( U ) ∶= ( ∑ x ∈ U ∣ g ( x )∣ p ) p and ∥ g ∥ L ∞ ( U ) ∶= sup x ∈ U ∣ g ( x )∣ . where the notation ∣ ⋅ ∣ denotes the euclidean norm on R k . Given a bounded subset U ⊆ Z d , we denote by L p ( U ) the normalized norms ∥ g ∥ L p ( U ) ∶= ( ∣ U ∣ ∑ x ∈ Z d ∣ g ( x )∣ p ) p . We introduce the normalized Sobolev norms H ( U ) and H − ( U ) by the formulas ∥ g ∥ H ( U ) ∶= U ∥ g ∥ L ( U ) + ∥∇ g ∥ L ( U ) and ∥ g ∥ H − ( U ) ∶= {( f, g ) U ∶ f ∶ U → R k , ∥ f ∥ H ( U ) ≤ } . We note that for each bounded connected subset U ⊆ Z d , the Poincar´e inequality implies the estimate ∥ g − ( g ) U ∥ L ( U ) ≤ C ( diam U ) ∥∇ g ∥ L ( U ) . We denote by H ( U ) the set of functions from U to R k which are equal to 0 outside the set U (by analogy tothe Sobolev space). We implicitly extend the functions of H ( U ) by the value 0 to the entire lattice Z d . For p, q ∈ [ , ∞] , we need to consider linear operators from L p ( Z d ) into L q ( Z d ) ; we introduce the operator normon this space according to the formula, for each A ∶ L p ( Z d ) → L q ( Z d ) , ∣∣∣ A ∣∣∣ L p ( Z d )→ L q ( Z d ) ∶= sup {∥ Au ∥ L q ( Z d ) ∶ u ∈ L p ( Z d ) , ∥ u ∥ L p ( Z d ) ≤ } . Note that, for each p ∈ [ , ∞] , the discrete gradient and Laplacian have a finite operator norm in the space L p ( Z d ) .We frequently consider functions defined from Z d an d valued in R of the form x → ∣ x ∣ − k . We implicitlyextend these functions at the point x = Z d . Forinstance, we may write, given two integers k, l ∈ N such that k + l > d and a point y ∈ Z d , ∑ x ∈ Z d ∣ x ∣ k ⋅ ∣ x − y ∣ l to mean ∑ x ∈ Z d ,x ∉{ ,y } ∣ x ∣ k ⋅ ∣ x − y ∣ l + ∣ y ∣ k + ∣ y ∣ l . Finally Section 4 of Chapter 8 requires to work with Fourier analysis, we thus introduce the Schwartz space ofrapidly decreasing functions of R d by the formula S ( R d ) ∶= { f ∈ C ∞ ( R d ) ∶ ∀ α, β ∈ N , sup x ∈ R d ∣ x ∣ α ∣∇ β f ∣ < ∞} as well as the set of tempered distributions S ′ ( R d ) to be its topological dual. Given a function g ∈ S ( R d ) , wedefine its Fourier transform ˆ g ∶ R d → R by the formulaˆ g ( ξ ) ∶= ∫ R d g ( x ) e − iξ ⋅ x d x, and extend the Fourier transform to the space of tempered distributions following the standard procedure. Given an pair of integers k, l ∈ N , we may identify thevector space R k × l with the space of ( k × l ) -matrices with real coefficients. Given a map F ∶ Z d → R k × l , wedenote its components by ( F ij ) ≤ i ≤ k, ≤ j ≤ l . For each integer i ∈ { , . . . , k } , we denote by F i ⋅ the map F i ⋅ ∶ ⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩ Z d → R l ,x ↦ ⎛⎝ k ∑ j = F ij ( x )⎞⎠ ≤ j ≤ l . We similarly define the map F ⋅ j for each integer j ∈ { , . . . , l } . We define the product between two maps F ∶ Z d → R l × k and g ∶ Z d → R k by the formula F g ∶ ⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩ Z d → R l ,x ↦ ⎛⎝ k ∑ j = F ij ( x ) f j ( x )⎞⎠ ≤ i ≤ l . We may abuse notations and write gF instead of F g . In Chapter 5, we need to study solutions of parabolicequations. We introduce in this section a few definitions and notations pertaining to this setting. For s > t ∈ R , we define the time intervals I s ∶= (− s, ] and I s ( t ) ∶= (− s + t, t ] . Given a point x ∈ Z d and a radius r >
0, we denote the parabolic cylinder by Q r ( t, x ) ∶= I r ( t ) × B ( x, r ) (where B ( x, r ) is the discrete ball). Tosimplify the notation, we write Q r to mean Q r ( , ) . Given a function u ∶ Q r ( t, x ) → R , we define its averageover the parabolic cylinder ( u ) Q r ( t,x ) by the formula ( u ) Q r ( t,x ) ∶= r ∣ B r ∣ ∫ − r ∑ x ∈ B r u ( t, x ) dt. Given a finite subset V ⊆ Z d or a bounded open set V ⊆ R d , we denote by ∂ ⊔ ( I r × V ) the parabolic boundaryof the cylinder I r × V defined by the formula ∂ ⊔ ( I r × V ) ∶= ( I r × ∂V ) ∪ ({− r } × V ) . Given a cube ◻ ⊆ Z d , we let Ω (◻) be the set ofvector-valued functions φ ∶ ◻ → R ( d ) such that φ = ∂ ◻ . We often drop the dependence on the domainwhen it is clear from the context. We also let Ω r be the set of vector-valued functions φ ∶ Z d → R ( d ) satisfyingthe growth condition ∑ x ∈ Z d ∣ φ ( x )∣ e −∣ x ∣ < ∞ . Given z ∈ Z d , we define τ z ∶ Ω → Ω to be a translation of elementsin Ω: τ z φ (⋅) = φ ( z + ⋅) .Given an inverse temperature β , a probability measure µ β on Ω and measurable function X ∶ Ω → R whichis either nonnegative or integrable with respect to the measure µ β , we denote its expectation and variance by ⟨ X ⟩ µ β ∶= ∫ Ω X ( φ ) µ β ( d φ ) and var µ β [ X ] = ∫ Ω ∣ X ( φ ) − ⟨ X ⟩ µ β ∣ µ β ( d φ ) . For each real number p ∈ [ , ∞) , we define the L p ( µ β ) -norm of the random variable X according to the formula ∥ X ∥ L p ( µ β ) ∶= (∫ Ω ∣ X ( φ )∣ p µ β ( d φ )) p and ∥ X ∥ L ∞ ( µ β ) ∶= ess sup φ ∈ Ω ∣ X ( φ )∣ . For each point x ∈ Z d and each integer i ∈ { , . . . , ( d )} , we let ω x,i be the function ω x,i ( y ) ∶= { e i if x = y x ≠ y, where ( e , . . . , e ( d ) ) is the canonical basis of R ( d ) . We define the differential operators ∂ x,i and ∂ x by theformulas ∂ x,i u ( φ ) ∶= lim h → u ( φ + hω x,i ) − u ( φ ) h and ∂ x u ( φ ) = ( ∂ x, u, . . . , ∂ x, ( d ) u ) . . NOTATIONS AND ASSUMPTIONS 17 We let C ∞ loc ( Ω ) be the set of smooth, local and compactly supported functions of the set Ω. We define thespace H ( µ β ) to be the closure of the space C ∞ loc ( Ω ) with respect to the norm (rescaled with respect to theinverse temperature β ) ∥ u ∥ H ( µ β ) ∶= ∥ u ∥ L ( µ β ) + ( β ∑ x ∈ Z d ∥ ∂ x u ∥ L ( µ β ) ) . For any subset U ⊆ Z d , we let L ( U, µ β ) to be the set of measurable functions u ∶ Z d × Ω → R k which satisfy ∥ u ∥ L ( U,µ β ) ∶= ( ∑ x ∈ U ∥ u ( x, ⋅)∥ L ( µ β ) ) < ∞ . When the set U is finite, we define the normalized L ( U, µ β ) -norm by the formula ∥ u ∥ L ( U,µ β ) ∶= ( ∣ U ∣ ∑ x ∈ U ∥ u ( x, ⋅)∥ L ( µ β ) ) , as well as the space-field average ( u ) U,µ β ∶= ∣ U ∣ ∑ x ∈ U ⟨ u ( x, ⋅)⟩ µ β . More generally, for p, q ∈ [ , ∞] and a set U ⊆ Z d , we introduce the L p ( U, L q ( µ β )) -norm by the formula ∥ u ∥ L p ( U,L q ( µ β )) ∶= ( ∑ x ∈ U ∥ u ( x, ⋅)∥ pL q ( µ β ) ) p . We define the norm H ( U, µ β ) by the formula ∥ u ∥ H ( U,µ β ) ∶= ( ∑ x ∈ U ∥ u ( x, ⋅)∥ H ( µ β ) + ∥∇ u ∥ L ( U,µ β ) ) , as well as the normalized H ( U, µ β ) -norm ∥ u ∥ H ( U,µ β ) ∶= ⎛⎝ ( diam U ) ∣ U ∣ ∑ x ∈ U ∥ u ( x, ⋅)∥ L ( µ β ) + β ∣ U ∣ ∑ x,y ∈ U ∥ ∂ y u ( x, ⋅)∥ L ( µ β ) + ∣ U ∣ ∥∇ u ∥ L ( U,µ β ) ⎞⎠ . We define the subset H ( U, µ β ) to be the subset of functions of H ( U, µ β ) which are equal to 0 on theboundary ∂U × Ω. We implicitly extend these functions by the value 0 to the space Z d . In particular, we alwaysthink of elements of H ( U, µ β ) as functions defined on the the entire space. We introduce the seminorm (cid:74) u (cid:75) H ( U,µ β ) ∶= ⎛⎝ β ∣ U ∣ ∑ x ∈ U,y ∈ Z d ∥ ∂ y u ( x, ⋅)∥ L ( µ β ) + ∣ U ∣ ∥∇ u ∥ L ( U,µ β ) ⎞⎠ . We define the H − ( U, µ ) -norm by the formula ∥ u ∥ H − ( U,µ β ) ∶= sup { ∣ U ∣ ∑ x ∈ U ⟨ u ( x, ⋅) v ( x, ⋅)⟩ µ β ∶ v ∈ H ( U, µ β ) , ∥ v ∥ H ( U,µ β ) ≤ } . We next state a Poincar´e inequality for H ( U, µ β ) . We give two statements, one for functions which vanishon the boundary of U and another for zero-mean functions in the case U is a cube. Lemma H ( U, µ β ) ) . Let ◻ L be a cube of size L . There exists C ( d, β ) < ∞ such that: (i) For every subset U ⊆ ◻ L and w ∈ H ( U, µ β ) , (1.7) ∥ w ∥ L ( U,µ β ) ≤ CL (cid:74) w (cid:75) H ( U,µ β ) . (ii) For every L ∈ N , every cube ◻ ′ ⊆ ◻ L and w ∈ H (◻ ′ , µ β ) , (1.8) ∥ w − ( w ) ◻ ′ ∥ L (◻ ′ ,µ β ) ≤ CL (cid:74) w (cid:75) H (◻ ′ ,µ β ) . Proof.
In the case of (i), since w vanishes on ∂U , the (discrete) Poincar´e inequality on U yields ⟨ ∑ x ∈ U ∣ w ( x, ⋅)∣ ⟩ µ β ≤ CL ∑ e ∈ E ( U ) ⟨∣∇ w ( e, ⋅)∣ ⟩ µ β . In the case of (ii), we may suppose without loss of generality that ( w ) ◻ ′ = ◻ ′ to obtain ⟨ ∑ x ∈◻ ′ w ( x, ⋅) ⟩ µ β ≤ ⟨ C diam (◻ ′ ) ∑ e ∈ E (◻ ′ ) (∇ w ( e, ⋅)) ⟩ µ β ≤ CL ∑ e ∈ E (◻ ′ ) ⟨(∇ w ( e, ⋅)) ⟩ µ β . (cid:3)
2. Discrete differential forms2.1. Definitions and basic properties.
Given an integer k ∈ { , . . . , d } , we denote by Λ k ( Z d ) the setof oriented k -cells of the hypercubic lattice Z d ;For each k -cell c k , we denote by c − k the same k -cell as c k with reverse orientation and by ∂c k the boundarythis cell. A k -form u is a mapping from Λ k (◻) to R such that u ( c − k ) = − u ( c k ) . Given a k -form u , we define its exterior derivative d u according to the formula, for each oriented ( k + ) -cell c k + ,(2.1) d u ( c k + ) = ∑ c k ⊆ ∂c k + u ( c k ) , where the orientation of the face c k is given by the orientation of the ( k + ) -cell c k + ; we set the conventiond u = d -form u . We define the codifferential d ∗ according to the formula, for each ( k − ) -cell c k − andeach k -form u ∶ Λ k (◻) → R ,(2.2) d ∗ u ( c k − ) ∶= ∑ ∂c k ∋ c k − u ( c k ) . Clearly, d u is a ( k + ) -form and d ∗ u is a ( k − ) -form; we set d ∗ u = u . One also verifies theproperties, for each k -form u ∶ Λ k (◻) → R , dd u = ∗ d ∗ u = . For arbitrary k -forms u, v ∶ Λ k ( Z d ) → R with finite support, we define the scalar product (⋅ , ⋅) by the formula(2.3) ( u, v ) = ∑ c k ∈ Λ k ( Z d ) u ( c k ) v ( c k ) . We restrict the scalar product (⋅ , ⋅) to forms which are only defined in a cube ◻ ; we denote the correspondingscalar product by (⋅ , ⋅) ◻ . It is defined by the formula, for each pair of forms k -forms u, v ∶ Λ k (◻) → R , ( u, v ) = ∑ c k ∈ Λ k (◻) u ( c k ) v ( c k ) . The codiferential d ∗ is the formal adjoint of the exterior derivative d with respect to this scalar product:Given a k -form u ∶ Λ k ( Z d ) → R and a ( k + ) -form v ∶ Λ k + ( Z d ) → R with finite supports, one has the identity(2.4) ( d u, v ) = ( u, d ∗ v ) . For an integer k ∈ { , . . . , d − } and a cube ◻ ⊆ Z d , we define the tangential boundary of the cube ∂ k, t ◻ tobe the set of all the k -cells which are included in the boundary of the cube ◻ . Given a k -form u ∶ Λ k (◻) → R ,we define its tangential trace t u to be the restriction of the form u to the set ∂ k, t ◻ . One has the formula, foreach k -form u ∶ Λ k (◻) → R such that t u = ( k + ) -form v ∶ Λ k (◻) → R , ( d u, v ) ◻ = ( u, d ∗ v ) ◻ . We will need the following lemma.
Lemma . Let ◻ ⊆ Z d be a cube of the lattice Z d of sidelength R and k be an integer in theset { , . . . , d − } . For each k -form f ∶ Λ k (◻) → R such that d f = and t f = on the tangential boundary ∂ k, t ◻ ,there exists a ( k − ) -form u ∶ Λ k − (◻) → R such that t u = on the tangential boundary ∂ k, t ◻ and d u = f inthe cube ◻ . Additionally, one can choose the form u such that R ∥ u ∥ L (◻) + ∥∇ u ∥ L (◻) ≤ C ∥ f ∥ L (◻) . . DISCRETE DIFFERENTIAL FORMS 19 An important role is played by the set of integer-valued, compactly supported forms q which satisfy d q = Q the set of these forms, i.e.,(2.5) Q ∶= { q ∶ Z d → Z ∶ ∣ supp q ∣ < ∞ , supp q is connected and d q = } . We may restrict our considerations to the charges of Q whose support is included in a cube ◻ ⊆ Z d ; to thisend, we introduce the notation Q ◻ ∶= { q ∶ Z d → Z ∶ supp q ⊆ ◻ , supp q is connected and d q = } . We will need to use the following version of Lemma 2.1 for the forms of the set Q . Lemma . Let k be an integer of the set { , . . . , d − } and q be a k -form with values in Z such that d q = , then there exists a ( k − ) -form n q with values in Z such that q = d n q .Moreover, n q can be chosen such that supp n q is contained in the smallest hypercube containing the support of q and such that ∥ n q ∥ L ∞ ≤ C ∥ q ∥ . As it is useful in the article, we record a series of inequalities satisfied by the charges q ∈ Q ,(2.6) ⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩ ∥ q ∥ L ∞ ≤ ∥ q ∥ , diam q ≤ ∣ supp q ∣ ≤ ∥ q ∥ , ∥ n q ∥ L ∞ ≤ C ∥ q ∥ , diam n q ≤ C ∥ q ∥ , ∣ supp n q ∣ ≤ C ∥ q ∥ d , ∥ n q ∥ L ≤ ∣ supp n q ∣ ∥ n q ∥ L ∞ ≤ C ∥ q ∥ d + , ∥ n q ∥ L ≤ ∥ n q ∥ L ∥ n q ∥ L ∞ ≤ C ∥ q ∥ d + . The proofs of these results use (2.2) and the fact that the charges are valued in the set Z ; they are left to thereader. Given a point ( x, y ) ∈ Z d × Z d , we denote by Q x and Q x,y the set of charges q ∈ Q such that the point x and the points x, y belong to the support of n q respectively, i.e., Q x ∶= { q ∈ Q ∶ x ∈ supp n q } and Q x,y ∶= { q ∈ Q ∶ x ∈ supp n q and y ∈ supp n q } . Similarly we also define Q ◻ ,x ∶= { q ∈ Q ◻ ∶ x ∈ supp n q } and Q ◻ ,x,y ∶= { q ∈ Q ◻ ∶ x ∈ supp n q and y ∈ supp n q } . We also record two inequalities involving the sum of charges: for each pair of points ( x, y ) ∈ Z d , each integer k ∈ N and each constants c > β ≥ ⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩ ∑ q ∈Q x ∥ q ∥ k e − c √ β ∥ q ∥ ≤ Ce − c √ β ∑ q ∈Q x,y ∥ q ∥ k e − c √ β ∥ q ∥ ≤ Ce − c √ β ∣ x − y ∣ where the constants C , c depend on k , c and the dimension d . The proofs of these inequalities rely on theassumption that the charges are integer-valued; they are fairly elementary and left to the reader. Given a subset I = ( i , . . . , i k ) ⊆ { , . . . , d } ofcardinality k . We denote by Λ kI ( Z d ) the set of oriented k -cells of the hypercubic lattice Z d which are parallelto the vectors ( e i , . . . , e i k ) . This set can be characterized as follows: if we let c I be the k -cell defined by theformula c I ∶= { k ∑ l = λ l e i l ∈ R d ∶ ≤ λ , . . . , λ k ≤ } , then we have(2.8) Λ kI ( Z d ) = { x + c I ∶ x ∈ Z d } . The identity (2.8) allows to identify the vector space of k -forms to the vector space of functions defined on Z d and valued in R ( dk ) according the procedure described below. Note that there are ( dk ) subsets of { , . . . , d } of cardinality k and consider an arbitrary enumeration I , . . . , I ( dk ) of these sets. To each k -form ˆ u ∶ Λ k ( Z d ) → R ,we can associate a vector-valued function u ∶ Z d → R ( dk ) defined by the formula, for each point x ∈ Z d ,(2.9) u ( x ) = ( ˆ u ( x + c I ) , . . . , ˆ u ( x + c I ( dk ) )) . This identification is enforced in most of the article; in fact, except in Section 1 of Chapter 3, we alwayswork with vector-valued functions instead of differential forms. We use the identification (2.9) to extend theformalism described in Section 1 to differential forms; we may for instance refer to the gradient of a form, orthe Laplacian of a form etc. Reciprocally, we extend the formalism described in Section 2.1 to vector-valuedfunctions; given a function u ∶ Z d → R ( dk ) , we may refer to the exterior derivative, the codifferential and thetangential trace of the function u , which we still denote by d u , d ∗ u and t u respectively. We note that thetwo definitions of the scalar products (1.3) for vector valued functions and (2.3) for differential forms coincidethrough the identification (2.9).From the definition of the exterior derivative d and the codifferential d ∗ given in (2.1) and (2.2) and theidentification (2.9), one sees that the differential operators d and d ∗ are linear functionals of the gradient ∇ :for each integer k ∈ { , . . . , d } , there exist linear maps L k, d ∶ R d ×( dk ) → R ( dk + ) and L k, d ∗ ∶ R d ×( dk ) → R ( dk − ) suchthat, for each function u ∶ Z d → R ( dk ) and each point x ∈ Z d ,(2.10) d u ( x ) = L k, d (∇ u ( x )) and d ∗ u ( x ) = L k, d ∗ (∇ u ( x )) . Using that linear maps on finite dimensional vector spaces are continuous, we obtain the estimates, for eachpoint x ∈ Z d , ∣ d u ( x )∣ ≤ C ∣∇ u ( x )∣ and ∣ d ∗ u ( x )∣ ≤ C ∣∇ u ( x )∣ , for some constant C depending only on the dimension d .This article frequently deals with functions defined on the space Z d × Ω × Z d (resp. Z d × Ω × Z d ) and valuedin R ( d )×( d ) (resp. R ( d ) ) since these maps correspond to the fundamental solutions of the Hellfer-Sj¨ostrandoperator (resp. differentiated Hellfer-Sj¨ostrand operator) associated to the dual Villain model introduced inSection 4.1 of Chapter 3. Given a map F ∶ Z d × Z d × Ω → R ( d )×( d ) , we denote by d x F ∶ Z d × Ω → R ( d )×( d ) ,d y F ∶ Z d × Ω → R ( d )×( d ) , d ∗ x F ∶ Z d × Ω → R d ×( d ) and d ∗ y F ∶ Z d × Ω → R ( d )× d the exterior derivative with respectto the first, second variable and the codifferential with respect to the first and second variable respectively.They are defined by the formulas, for each triplet ( x, y, φ ) ∈ Z d × Z d × Ω and each integer k ∈ { , . . . , ( d )} , ( d x F ( x, y, φ )) ⋅ k = L , d (∇ x F ⋅ k ( x, y, φ )) , ( d y F ( x, y, φ )) k ⋅ = L , d (∇ y F k ⋅ ( x, y, φ )) and ( d ∗ x F ( x, y, φ )) ⋅ k = L , d ∗ (∇ x F ⋅ k ( x, y, φ )) , ( d ∗ y F ( x, y, φ )) k ⋅ = L , d ∗ (∇ y F k ⋅ ( x, y, φ )) . Similarly, given a function F ∶ Z d × Ω × Z d → R ( d ) , we define, for each ( x, y, φ, x , y ) ∈ Z d × Z d × Ω × Z d × Z d ,each field φ ∈ Ω and each triplet of integers i, j, k ∈ { , . . . , ( d )}⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩ ( d x F ( x, y, φ, x , y )) ⋅ ijk = L , d (∇ x F ⋅ ijk ( x, y, φ, x , y )) , ( d y F ( x, y, φ, x , y )) i ⋅ jk = L , d (∇ x F i ⋅ jk ( x, y, φ, x , y )) , ( d x F ( x, y, φ, x , y )) ij ⋅ k = L , d (∇ x F ij ⋅ k ( x, y, φ, x , y )) , ( d y F ( x, y, φ, x , y )) ijk ⋅ = L , d (∇ x F ijk ⋅ ( x, y, φ, x , y )) , and similarly ⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩ ( d ∗ x F ( x, y, φ, x , y )) ⋅ ijk = L , d ∗ (∇ x F ⋅ ijk ( x, y, φ, x , y )) , ( d ∗ y F ( x, y, φ, x , y )) i ⋅ jk = L , d ∗ (∇ x F i ⋅ jk ( x, y, φ, x , y )) , ( d ∗ x F ( x, y, φ, x , y )) ij ⋅ k = L , d ∗ (∇ x F ij ⋅ k ( x, y, φ, x , y )) , ( d ∗ y F ( x, y, φ, x , y )) ijk ⋅ = L , d ∗ (∇ x F ijk ⋅ ( x, y, φ, x , y )) . We extend these definitions so that we can consider mixed derivatives ; for instance, we may use the notationd ∗ y d ∗ x F (or any other combination of exterior derivatives and codifferentials). It is clear that as long as thederivatives involve different variables, they commute: we have for instance d ∗ y d ∗ x F = d ∗ x d ∗ y F . . DISCRETE DIFFERENTIAL FORMS 21 We record the following identity which relates the Laplacian ∆ to the exterior derivative d and thecodifferential d ∗ ,(2.11) − ∆ = dd ∗ + d ∗ d . Using the identity d ○ d =
0, one obtains that the Laplacian commutes with the exterior derivative: we haved∆ = − d ( dd ∗ + d ∗ d ) = − dd ∗ d = − ( dd ∗ d + d ∗ d ) d = ∆d . Similarly, using this time the identity d ∗ ○ d ∗ =
0, we obtain that the Laplacian commutes with the codifferential:we have d ∗ ∆ = − d ∗ ( dd ∗ + d ∗ d ) = − d ∗ dd ∗ = − ( dd ∗ d + d ∗ d ) d ∗ = ∆d ∗ . We complete this section by recording the Gaffney-Friedrichs inequality which provides an upper bound onthe L -norm of the gradient of a form in terms of the L -norm of its exterior derivative and the codifferentialassuming that the tangential trace of the form vanishes. Proposition . Let ◻ be a cube of Z d . Then there exists aconstant C ∶= C ( d ) < ∞ such that for each k -form u ∶ Λ k (◻) → R with vanishing tangential trace, we have ∥∇ u ∥ L (◻) ≤ C (∥ d u ∥ L (◻) + ∥ d ∗ u ∥ L (◻) ) . The proof of the continuous version of this inequality can be found in [
39, 33 ] or in the monograph [ ,Proposition 2.2.3]. We complete this section by proving the solvability of a boundary value problem involvingdiscrete differential forms used in Section 1 of Chapter 3. Proposition . For any integer k ∈ { , . . . , d − } and any cube ◻ ∈ Z d and any k -form q ∶= ( q , . . . , q ( dk ) ) ∶◻ → R ( dk ) such that d q = in the cube ◻ and t q = on the boundary ∂ ◻ , there exists a unique solution to theboundary value problem (2.12) ⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩ dd ∗ w = q in ◻ , d w = in ◻ , t w = on ∂ ◻ , t d ∗ w = on ∂ ◻ . If we denote by w , . . . , w ( dk ) the coordinates of the map w , then they solve the following boundary value problem:for each i ∈ { , . . . , ( dk )} , if we denote by ∂ I i ◻ the subset of faces of the boundary ∂ ◻ which are parallel to thecell c I i , then we have (2.13) ⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩ − ∆ w i = q i in ◻ ,w i = in ∂ I i ◻ , ∇ w i ⋅ n = on ∂ ◻ ∖ ∂ I i ◻ . Remark . The boundary condition (2.13) is a combination of the Dirichlet and Neumann boundaryconditions: given an integer i ∈ { , . . . , ( dk )} , we assign Dirichlet boundary condition on the faces which areparallel to the cell c I i and Neumann boundary condition on the faces which are orthogonal to the cell c I i . Proof.
The boundary value problem (2.12) admits a variational formulation which we can use to proveexistence and uniqueness of solutions. We first define the set of k -forms C k (◻) ∶= { u ∶ ◻ → R ( dk ) ∶ d u = ◻ and t u = ∂ ◻} . We then define the energy functional J q ∶ C k (◻) → R according to the formula J q ( u ) ∶= ∥ d ∗ u ∥ L (◻) − ( q, u ) ◻ . To prove the solvability of the problem (2.12), we prove that there exists unique minimizer to the variationalproblem inf u ∈ C k (◻) J ( u ) . We first use that, by Lemma 2.1, there exists a ( k − ) -form n q ∶ ◻ → R ( dk − ) such that t n q = ∂ ◻ andd n q = q in the cube ◻ . We then perform an integration by parts to write J q ( u ) = ∥ d ∗ u ∥ L (◻) − ( n q , d ∗ u ) ◻ . The technique then follows the standard strategy of the calculus of variations. The energy functional J q isbounded from below and we consider a minimizing sequence ( w n ) n ∈ N . It is clear that the norms ∥ d ∗ w n ∥ L (◻) areuniformly bounded in n ∈ N . Using that d w n = ∥∇ w n ∥ L (◻) and ∥ w n ∥ L (◻) are uniformly bounded in n . We can thus extracta subsequence which converges in the discrete space L (◻) and verify that the limit is solution to theproblem (2.12). The uniqueness is a consequence of the uniform convexity of the functional J q .To prove (2.13), note that the condition d w = − ∆ w = q in the cube ◻ .Using the definition of the Laplacian for vector-valued function (stated in (1.2)), we have that for each integer i ∈ { , . . . , ( dk )} , − ∆ w i = q i in the cube ◻ . The boundary condition t w = w i is equal to 0 on eachface which is parallel to the cell c I i ; the condition t d ∗ w = w i satisfies a Neumannboundary condition on the faces of the boundary ∂ ◻ which are orthogonal to the cell c I i . (cid:3)
3. Convention for constants and exponents
Throughout this article, the symbols c and C denote positive constants which may vary from line to line.These constants may depend only on the dimension d and the inverse temperature β . We use the symbols α, β, γ, δ to denote positive exponents which depend only on the dimension d . Usually, we use the letter C for large constants (whose value is expected to belong to [ , ∞) ) and c for small constants (whose value isexpected to be in ( , ] ). The values of the exponents α, β, γ, δ are always expected to be small. When theconstants and exponents depend on other parameters, we write it explicitly and use the notation C ∶= C ( d, β, t ) to mean that the constant C depends on the parameters d, β and t .When the constants depend on the charges q ∈ Q (see (2.5)), we frequently need to keep track of theirdependence in this parameter; more specifically we need that the growth of the constant C is at most algebraicin the parameter ∥ q ∥ . We usually denote by C q a constant which depends on the parameter d, β and q andwhich satisfies the growth condition C q ≤ C ∥ q ∥ k , for some C ∶= C ( d, β ) < ∞ and k ∶= k ( d ) < ∞ . We allow thevalues of C and k to vary from line to line and we may write C q + C q ≤ C q or C q C q ≤ C q . We usually do not keep track of the dependence of the constants in the inverse temperature β (even thoughwe believe it should be possible with our techniques) except in Chapters 5 and 6. In these two chapters, weassume that the constants depend only on the dimension d and make it explicit if they depend on the inversetemperature β .HAPTER 3 Duality and Helffer-Sj¨ostrand representation
1. From Villain model to solid on solid model
In this section we recall the duality relation between the Villain model in Z d and a statistical mechanicalmodel of lattice Coulomb gas, with integer valued and locally neutral charges (which can also be viewed asa solid-on-solid model) defined on Λ ( Z d ) , as observed in [ ]. One may then perform a Fourier transformwith respect to the charge variable, and obtain a classical random field representation of the Coulomb gas,known as the sine-Gordon representation. When the temperature is low enough, we may apply a one-steprenormalization argument, following the presentation of Bauerschmidt [ ] (see also [ ]), to reduce theeffective activity of the charges, thus obtain an effective, real valued random interface model on 2-forms with aconvex action.Recall that the partition function for the Villain model in a cube ◻ ⊆ Z d with zero boundary condition isgiven by Z ◻ , ∶= ∫ ∏ e ⊆ E (◻) ∑ m ∈ Z exp (− β (∇ θ ( e ) − πm ) ) ∏ x ∈ ∂ ◻ δ ( θ ( x )) ∏ x ∈◻ ○ [− π,π ) ( θ ( x )) d θ ( x ) . Since we need to use the formalism of discrete differential forms later in this chapter, we note that the function θ ∶ ◻ ↦ R can be seen as a 0-form, in that case the discrete gradient ∇ θ can be seen as a 1-form and is equal tothe exterior derivative d θ . We may thus rewrite Z ◻ , ∶= ∫ ∏ e ⊆ E (◻) ∑ m ∈ Z exp (− β ( d θ ( e ) − πm ) ) ∏ x ∈ ∂ ◻ δ ( θ ( x )) ∏ x ∈◻ ○ [− π,π ) ( θ ( x )) d θ ( x ) . Permuting the sum with the product and the integral, we obtain(1.1) Z ◻ , = ∑ m ∈ Z E (◻) t = ∫ ∏ e ⊆ E (◻) exp (− β ( d θ ( e ) − π m ( e )) ) ∏ x ∈ ∂ ◻ δ ( θ ( x )) ∏ x ∈◻ ○ [− π,π ) ( θ ( x )) d θ ( x ) , where we have used the notation Z E (◻) t = ∶= { m ∶ E (◻) ↦ Z ∶ tm = ∂ ◻} . Observe that we may split the sum according to(1.2) ∑ m ∈ Z E (◻) t = = ∑ q ∈ Z F (◻) t = , d q = ∑ m ∈ Z E (◻) t = , d m = q , where we have set Z F (◻) t = ∶= { q ∶ F (◻) ↦ Z ∶ t q = ∂ ◻} . A combination of (1.1) and (1.2) yields Z ◻ , = ∑ q ∈ Z F (◻) t = , d q = ∑ m ∈ Z E (◻) t = , d m = q ∫ ∏ e ⊆ E (◻) exp (− β ( d θ ( e ) − π m ( e )) ) ∏ x ∈ ∂ ◻ δ ( θ ( x )) ∏ x ∈◻ ○ [− π,π ) ( θ ( x )) d θ ( x ) . Here q ∶ F (◻) → Z is the “vortex charge” on each plaquette of ◻ , which arises, informally, from ∮ F d θ ( e ) = πq ( F ) . For each q ∈ Z F (◻) t = , satisfying d q =
0, we denote by n q an element of Z E (◻) t = such that d n q = q , chosen arbitrarilyamong all the possible candidates, note that the set of candidates is not empty by Proposition 2.2 of Chapter 2.
234 3. DUALITY AND HELFFER-SJ ¨OSTRAND REPRESENTATION
Using that each 1-form m ∈ Z E (◻) t = satisfying d m = w , for some w ∶ ◻ ↦ Z satisfying w = ∂ ◻ , one can rewrite the previous display according to Z ◻ , = ∑ q ∈ Z F (◻) t = , d q = ∑ w ∈ Z ◻ ∫ ∏ e ⊆ E (◻) exp (− β ( d θ ( e ) − π ( n q + d w ) ( e )) ) ∏ x ∈ ∂ ◻ δ ( θ ( x )) ∏ x ∈◻ ○ [− π,π ) ( θ ( x )) dθ ( x ) , where we have set Z ◻ ∶= { w ∶ ◻ ↦ Z ∶ w = ∂ ◻} . Using the change of variable φ ∶= θ + πw , and summing over all the maps w ∈ Z ◻ , one obtains Z ◻ , = ∑ q ∈ Z F (◻) t = , d q = ∫ R ◻ ∏ e ⊆ E (◻) exp (− β ( d φ ( e ) − π n q ( e )) ) ∏ x ∈ ∂ ◻ δ ( φ ( x )) ∏ x ∈◻ ○ dφ ( x ) . Using Proposition 2.4 of Chapter 2, we denote by ( dd ∗ ) − q the (unique) solution of the problem ⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩ dd ∗ w = q in ◻ , d w = ◻ , t w = ∂ ◻ , t d ∗ w = ∂ ◻ . It is then clear that the charge n q − d ∗ ( dd ∗ ) − q satisfies d ( n q − d ∗ ( dd ∗ ) − q ) = q − q = ◻ and t ( n q − d ∗ ( dd ∗ ) − q ) = ∂ ◻ . Applying Proposition 2.1 of Chapter 2, one can write(1.3) n q = d φ n q + d ∗ ( dd ∗ ) − q, for some φ n q ∶ ◻ ↦ R with φ n q = ∂ ◻ . As a remark, note that the operator ( dd ∗ ) − depends on the cube ◻ ,since this cube is fixed through the proof, we omit the dependence in the cube ◻ in the notation. Then usingthe translation invariance of the Lebesgue measure, we obtain Z ◻ , = ∑ q ∈ Z F (◻) t = , d q = ∫ R ◻ ∏ e ⊆ E (◻) exp (− β ( d φ ( e ) − π d ∗ ( dd ∗ ) − q ( e )) ) ∏ x ∈ ∂ ◻ δ ( φ ( x )) ∏ x ∈◻ d φ ( x ) . The previous identity can be simplified Z ◻ , = Z GF F × Z ( ) (1.4) ∶= ∫ R ◻ exp (− β ( d φ, d φ )) ∏ x ∈ ∂ ◻ δ ( φ ( x )) ∏ x ∈◻ d φ ( x ) × ∑ q ∈ Z F (◻) t = , d q = exp (− π β ( q, ( dd ∗ ) − q )) . Using the identity d φ = ∇ φ (valid for 0-forms), we see that the first term in the left hand side of (1.4) is thepartition function of the discrete Gaussian free field in the cube ◻ with Dirichlet boundary condition. In otherwords, the Villain partition function factorizes into the partition function of a Gaussian free field in F (◻) , andthe vortex charges that form a (neutral) Coulomb gas.One can use the same argument to study the two-point functions ⟨ e i ( θ ( x )− θ ( )) ⟩ µ Vβ, ◻ , , For any point x ∈ ◻ , consider the string observable h ,x ∶ E (◻) ↦ Z be such that d ∗ h ,x = x − and h x ∶ E (◻) ↦ Z such that d ∗ h x = x , with the same computation, we obtain(1.5) ⟨ e i ( θ ( x )− θ ( )) ⟩ µ Vβ, ◻ , = ⟨ e i ( φ ( x )− φ ( )) ⟩ GF F ⟨ e − iπ ( q, ( dd ∗ ) − d h ,x ) ⟩ µ C ( β ) and ⟨ e iθ ( x ) ⟩ µ Vβ, ◻ , = ⟨ e iφ ( x ) ⟩ GF F ⟨ e − iπ ( q, ( dd ∗ ) − d h x ) ⟩ µ C ( β ) . Here ⟨ e i ( φ ( x )− φ ( )) ⟩ GF F ∶= Z − GF F × ∫ R ◻ e i ( φ ( x )− φ ( )) exp (− β (∇ φ, ∇ φ )) ∏ x ∈ ∂ ◻ δ ( φ ( x )) ∏ x ∈◻ ○ dφ ( x ) and ⟨ e − iπ ( q, ( dd ∗ ) − d h ,x ) ⟩ µ C ( β ) = Z ( ) − × ∑ q ∈ Z F (◻) t = , d q = e − π β ( q, ( dd ∗ ) − q ) e − iπ ( q, ( dd ∗ ) − d h ,x ) . . FROM VILLAIN MODEL TO SOLID ON SOLID MODEL 25 Following [ ], we define the notations σ x ∶= ( dd ∗ ) − d h x and σ x ∶= ( dd ∗ ) − d h ,x . Note that for each q ∈ Z E (◻) t = satisfying d q =
0, one has(1.6) ( q, σ x − σ ) = ( q, σ x ) mod Z . To justify the identity (1.6), we note that, with the same argument as in (1.3), we may write h x − h − h x = d φ + d ∗ σ x − d ∗ σ − d ∗ σ x , for some field φ ∶ ◻ ↦ R satisfying φ = ∂ ◻ . Taking the scalar product with the 1-form d φ and performing integrations by parts, we obtain that ( d φ, h x − h − h x ) = ( φ, d ∗ h x − d ∗ h − d ∗ h x ) = ( φ, x − − ( x − )) = ( d φ + d ∗ σ x − d ∗ σ − d ∗ σ x , d φ ) = ( d φ, d φ ) + ( σ x − σ − σ x , dd φ ) = ( d φ, d φ ) . A combination of the two previous displays implies d φ = h x − h − h x = d ∗ σ x − d ∗ σ − d ∗ σ x . Wethen use that q = d n q for some n q ∈ Z E (◻) t = to write ( q, σ x − σ − σ x ) = ( n q , d ∗ σ x − d ∗ σ − d ∗ σ x ) = ( n q , h x − h − h x ) ∈ Z . This is (1.6). A consequence of (1.6) is that for each q ∈ Z E (◻) t = satisfying d q = e − iπ ( q,σ x − σ ) = e − iπ ( q,σ x ) . For later use, we note that, by Proposition 2.4 of Chapter 2, the maps σ , σ x and σ x can be equivalentlydefined by the formulas σ x = (− ∆ ) − d h x , σ = (− ∆ ) − d h x and σ = (− ∆ ) − d h x in ◻ where the Laplacian is subject to the boundary condition stated in (2.13). In particular, using that theLaplacian commutes with the operators d and d ∗ , we formally obtain(1.7) d ∗ σ = (− ∆ ) − d ∗ d h x = − h x − (− ∆ ) − dd ∗ h = − h − (− ∆ ) − d = − h − ∇ G, where G is the standard random walk Green’s function on the lattice Z d and where we have used the identityd = ∇ valid for any function defined on Z d . A consequence of the identity (1.7) is the equality(1.8) e − iπ ( q,σ ) = e − iπ ( n q , ∇ G ) . While the identity (1.7) is not exactly true in finite volume (since we cannot a priori commute the operators d,d ∗ and (− ∆ ) − ), it becomes true by taking the infinite volume limit (i.e., sending the volume of the cube ◻ to ∞ ). To avoid further technicalities, we will assume that the identity (1.8) holds in the rest of this chapter.Similar statements hold for the maps σ x and σ x and we may write e − iπ ( q,σ x ) = e − iπ ( n q , ∇ G x ) and e − iπ ( q,σ x ) = e − iπ ( n q , ∇ G x −∇ G ) . where we have used the notation G x ∶= G (⋅ − x ) . We set the notation, for each σ ∶ F (◻) ↦ R ,(1.9) Z ( σ ) ∶= ∑ q ∈ Z F (◻) t = , d q = e − π β ( q, ( dd ∗ ) − q ) e − iπ ( q,σ ) . So that Z ( σ x ) Z ( ) = ⟨ e − iπ ( q,σ x ) ⟩ µ C ( β ) and Z ( σ x ) Z ( ) = ⟨ e − iπ ( q,σ x ) ⟩ µ C ( β ) . Let φ , . . . , φ ( d ) be independent, real-valued, Gaussian free fields in the cube ◻ with boundary conditions givenby (2.13) of Chapter 2. We denote by φ ∶= ( φ , . . . , φ ( dk ) ) the corresponding vector-valued Gaussian field, it isvalued in the space C (◻) ∶= { w ∶= ( w , . . . , w ( d ) ) ∶ ◻ → R ( d ) ∶ ∀ i ∈ { , . . . , ( d )} , w i = ∂ ◻ ∖ ∂ I i ◻} . By Proposition 2.4 of Chapter 2, if we denote by (− ∆ ) − q the solution of the boundary value problem (2.13),then one has the identity (− ∆ ) − q = ( dd ∗ ) − q . This observation implies that for each q ∈ Z E (◻) satisfyingd q = t q = E [ e iπ ( q,φ ) ] = e − π β ( q, (− ∆ ) − q ) = e − π β ( q, ( dd ∗ ) − q ) . Consequently, Z ( σ x ) = ∑ q ∈ Z F (◻) t = , d q = E [ e − iπ ( q,φ + σ x ) ] . Thus the partition function of this lattice Coulomb gas can be represented in terms of a characteristic functionwith respect to a Gaussian measure. We then claim that for β sufficiently large, a one-step renormalizationmaps the Coulomb gas model to an effective one with very small effective activity. Using that the discreteLaplacian is bounded from above, one has that (− ∆ ) − ≥ c , for some c ∶= c ( d ) >
0. We then choose the inversetemperature β larger than then value c and decompose the Gaussian field φ as the sum of two independentGaussian fields φ + φ , such that φ and φ have covariance matrices β ((− ∆ ) − − β − Id ) and β Id . We canthus write Z ( σ x ) = ∑ q ∈ Z F (◻) t = , d q = E [ e − iπ ( q,φ + φ + σ x ) ] = ∑ q ∈ Z F (◻) t = , d q = e − π β / ( q,q ) E µ [ e − iπ ( q,φ + σ x ) ] , where µ is a Gaussian measure on C (◻) , given by dµ ( φ ) = Const × exp (− ( φ , β ((− ∆ ) − − β − Id ) − φ )) φ ∈ C (◻) dφ . For β sufficiently large, we may expand ((− ∆ ) − − β − Id ) − into a convergent sum ((− ∆ ) − − β − Id ) − = − ∆ + ∑ n ≥ β n / (− ∆ ) n + . Thus dµ ( φ ) = Z − × exp ( β ( φ , ∆ φ ) − ∑ n ≥ β β n / ( φ , (− ∆ ) n + φ )) φ ∈ C (◻) dφ . Following [ ], (especially see Lemmas 5.14 and 5.15 there), since e − π β / ( q,q ) decays to zero rapidly in ∥ q ∥ ∶= ∑ x ∈ F (◻) ∣ q ( x )∣ , we may apply a standard cluster expansion to conclude that for β large enough, one canre-sum Z ( σ x ) as Z ( σ x ) = E µ ⎡⎢⎢⎢⎢⎣ exp ⎛⎝ ∑ q ∈Q ◻ z ( β, q ) e − iπ ( q,φ + σ x ) ⎞⎠⎤⎥⎥⎥⎥⎦ = E µ ⎡⎢⎢⎢⎢⎣ exp ⎛⎝ ∑ q ∈Q ◻ z ( β, q ) cos 2 π ( q, φ + σ x )⎞⎠⎤⎥⎥⎥⎥⎦ , where the sum is over all lattice animals q ∈ Q ◻ with connected support satisfying d q = t q = z ( β, q ) is a real number satisfying the estimate(1.10) ∣ z ( β, q )∣ ≤ e − cβ / ∥ q ∥ , for some c ∶= c ( d ) > . Similarly, Z ( ) = E µ ⎡⎢⎢⎢⎢⎣ exp ⎛⎝ ∑ q ∈Q ◻ z ( β, q ) e − iπ ( q,φ ) ⎞⎠⎤⎥⎥⎥⎥⎦ = E µ ⎡⎢⎢⎢⎢⎣ exp ⎛⎝ ∑ q ∈Q ◻ z ( β, q ) cos 2 π ( q, φ )⎞⎠⎤⎥⎥⎥⎥⎦ . Using the trigonometric identitycos 2 π ( q, φ + σ x ) = cos 2 π ( q, φ ) cos 2 π ( q, σ x ) − sin 2 π ( q, φ ) sin 2 π ( q, σ x ) , we may write(1.11) Z ( σ x ) Z ( ) = ⟨ exp ⎛⎝ ∑ q ∈Q ◻ z ( β, q ) sin 2 π ( φ, q ) sin 2 π ( σ x , q ) + ∑ q ∈Q ◻ z ( β, q ) cos 2 π ( φ, q ) ( cos 2 π ( σ x , q ) − )⎞⎠⟩ µ β, ◻ . Here µ β, ◻ is defined as a measure on the space C (◻) by dµ β, ◻ ( φ ) ∶= Const × exp ⎛⎝ β ( φ, ∆ φ ) − ∑ n ≥ β β n / ( φ, (− ∆ ) n + φ ) + ∑ q ∈Q ◻ z ( β, q ) cos 2 π ( q, φ )⎞⎠ φ ∈ C (◻) d φ. . FROM VILLAIN MODEL TO SOLID ON SOLID MODEL 27 Combining (1.5) and (1.11), we have the following dual representation for the two-point function of theVillain model. Define G ◻ be the solution of the problem(1.12) { − ∆ G ◻ ( x, ⋅) = δ x in ◻ ,G ◻ ( x, ⋅) = ∂ ◻ . Proposition . Let G ◻ be defined as above. For β sufficiently large, we have (1.13) ⟨ e i ( θ ( x )− θ ( )) ⟩ µ Vβ, ◻ , exp ( β G ◻ ( , x ))= ⟨ exp ⎛⎝ ∑ q ∈Q ◻ z ( β, q ) sin 2 π ( φ, q ) sin 2 π ( σ x , q ) + ∑ q ∈Q ◻ z ( β, q ) cos 2 π ( φ, q ) ( cos 2 π ( σ x , q ) − )⎞⎠⟩ µ β, ◻ . Following the same argument we also obtain the dual representation for ⟨ e i ( θ ( x )+ θ ( )) ⟩ µ Vβ, ◻ , . Define σ x ∶= σ + σ x . We then have(1.14) ⟨ e i ( θ ( x )+ θ ( )) ⟩ µ Vβ, ◻ , exp ( β G ◻ ( , x ))= ⟨ exp ⎛⎝ ∑ q ∈Q ◻ z ( β, q ) sin 2 π ( φ, q ) sin 2 π ( σ x , q ) + ∑ q ∈Q ◻ z ( β, q ) cos 2 π ( φ, q ) ( cos 2 π ( σ x , q ) − )⎞⎠⟩ µ β, ◻ . In view of (1.5), to study the two-point function of the (finite volume) Villain model, it suffices to computethe expectation of a nonlinear functional (1.13) with respect to the Gibbs measure µ β, ◻ . Notice that for β large, the exponential smallness of z ( β, q ) implies that µ β, ◻ is a perturbation of a Gaussian measure. Theneutrality condition d q = µ β, ◻ is a measure of gradient-type , i.e., the Hamiltonian only dependson d φ . In later sections, we combine the Helffer-Sj¨ostrand representation with quantitative homogenizationto show that, for sufficiently large β , on large scales (i.e., ∣ x ∣ → ∞ and ∣ ◻ ∣ → ∞ ) the measure µ β, ◻ behaveslike an effective Gaussian free field, with the covariance matrix depending on β . This shows, along the line ofthe Gaussian heuristics, that the subleading order of (1.11) (and therefore, the truncated two-point function)decays asymptotically as C ∣ x ∣ − d where the constant depends on β . Remark . To see that the Gaussian heuristics implies that for β sufficiently large, (1.13) has anasymptotic power law ∣ x ∣ − d , we begin by noting that the assumption (1.10) implies that for β ≫
1, charges in Q ◻ are essentially supported on dipoles, i.e., of the form q = z ( β )( δ x − δ x + e i ) , for i = , . . . , d . Thus the rightside of (1.13) is approximately ⟨ exp ⎛⎝ ∑ e ∈ E (◻) z ( β ) sin 2 π (∇ φ ( e )) sin 2 π (∇ G ( e ) − ∇ G x ( e ))⎞⎠× exp ⎛⎝ ∑ e ∈ E (◻) z ( β ) cos 2 π (∇ φ ( e )) ( cos 2 π (∇ G ( e ) − ∇ G x ( e )) − )⎞⎠⟩ µ β, ◻ . Since ∣ cos 2 π (∇ G ( e ) − ∇ G x ( e )) − ∣ ≤ C (∇ G ( e ) − ∇ G x ( e )) decays fast away from 0 and x , let us assumefor now that the term ∑ e ∈ E (◻) z ( β ) cos 2 π (∇ φ ( e )) ( cos 2 π (∇ G ( e ) − ∇ G x ( e )) − ) only contributes to the lowerorder. By further making the approximation sin a ≈ a for small a , we may further approximate the expressionabove by ⟨ exp ⎛⎝ ∑ e ∈ E (◻) z ( β ) π (∇ φ ( e )) π (∇ G ( e ) − ∇ G x ( e ))⎞⎠⟩ µ β, ◻ . Using an integration by parts, this equals to ⟨ exp ( π ( φ ( ) − φ ( x )))⟩ µ β, ◻ . Note that for β sufficiently large, µ β, ◻ is a small perturbation of a Gaussian free field, we may conclude ⟨ exp ( π ( φ ( ) − φ ( x )))⟩ µ β, ◻ ≈ exp (
12 var µ β, ◻ ( π ( φ ( ) − φ ( x )))) ≈ C ( d, β ) + C ( d, β )∣ x ∣ − d . We remark that the computation above is only heuristical and the constants C , C obtained are not theright constants. Indeed, the nonlocal charges in Q ◻ , the nonlinear function sin x , and the non-Gaussian field µ β, ◻ contribute to a nontrivial correction of these constants. Such corrections can be obtained rigorouslythrough the homogenization of the Helffer-Sj¨ostrand PDE. Remark . The preceding derivation is for the measure with Dirichlet boundary condition. For theVillain model with Neumann and periodic boundary conditions, similar dual representation holds. To statethe result, we fix a base point x ∗ ∈ ∂ ◻ or x ∗ ∈ T . Define dµ β, ◻ ,x ∗ ( φ ) ∶= Const × exp ⎛⎝ β ( φ, ∆ φ ) − ∑ n ≥ β β n / ( φ, (− ∆ ) n + φ ) + ∑ q ∈Q ◻ z ( β, q ) cos 2 π ( q, φ )⎞⎠ φ ( x ∗ )= dφ and dµ β, T ,x ∗ ( φ ) ∶= Const × exp ⎛⎝ β ( φ, ∆ φ ) T − ∑ n ≥ β β n / ( φ, (− ∆ ) n + φ ) T + ∑ q ∈Q ◻ z ( β, q ) cos 2 π ( q, φ )⎞⎠ φ ( x ∗ )= dφ We then have ⟨ e i ( θ ( x )− θ ( )) ⟩ µ Vβ, ◻ , f exp ( β G ◻ ,x ∗ ( , x ))= ⟨ exp ⎛⎝ ∑ q ∈Q ◻ z ( β, q ) sin 2 π ( φ, q ) sin 2 π ( σ x , q ) + ∑ q ∈Q ◻ z ( β, q ) cos 2 π ( φ, q ) ( cos 2 π ( σ x , q ) − )⎞⎠⟩ µ β, ◻ ,x ∗ and ⟨ e i ( θ ( x )− θ ( )) ⟩ µ Vβ, T exp ( β G T ,x ∗ ( , x ))= ⟨ exp ⎛⎝ ∑ q ∈Q ◻ z ( β, q ) sin 2 π ( φ, q ) sin 2 π ( σ x , q ) + ∑ q ∈Q ◻ z ( β, q ) cos 2 π ( φ, q ) ( cos 2 π ( σ x , q ) − )⎞⎠⟩ µ β, T ,x ∗ , where G ◻ ,x ∗ (and G T ,x ∗ ) are Green’s function in ◻ (and T ), with zero boundary condition at x ∗ , defined by ⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩ − ∆ G ◻ ,x ∗ ( x, ⋅) = δ x − δ x ∗ in ◻ ,G ◻ ,x ∗ ( x, x ∗ ) = , ∇ G ◻ ,x ∗ ( x, ⋅) ⋅ n = ∂ ◻ and { − ∆ G T ,x ∗ ( x, ⋅) = δ x − δ x ∗ in T ,G T ,x ∗ ( x, x ∗ ) = .
2. Brascamp-Lieb inequality
For L ≥ ◻ L ∶= [− L, L ] d ∩ Z d and write µ β, ◻ L as µ β,L . As we discussed, when β is sufficientlylarge, the measure µ β,L is a small perturbation of the Gaussian measure, and is therefore log-concave. We nextpresent the Brascamp-Lieb inequality [
17, 16 ], which states that the variance of observables with respect to alog-concave measure is dominated by that of a Gaussian measure. Denote, for any x ∈ Z d , ∂ x ∶= ( ∂ x, , ⋯ , ∂ x, ( d ) ) and ∂ ∶= ( ∂ x ) x ∈ Z d . We let G C (◻) be the Green’s function defined by the formula { − ∆ G C (◻) ( x, ⋅) = δ x in ◻ G C (◻) ( x, ⋅) ∈ C (◻) , we denote its components by G C (◻) , 1 ≤ i ≤ ( d ) . Proposition µ β, ◻ ) . Let β be sufficiently large. For every F ∈ H ( µ β, ◻ ) ,there exists C = C ( d, β ) < ∞ such that (2.1) var µ β, ◻ [ F ] ≤ C ∑ x,y ∈◻ ○ ( d ) ∑ i = G C (◻) ,i ( x, y ) ⟨( ∂ x,i F ) ( ∂ y,i F )⟩ µ β, ◻ . Given f ∶ ◻ → R ( d ) , recall that we denote by ( f, φ ) ∶= ∑ x ∈◻ f ( x ) φ ( x ) the linear functional of φ . Here wefollow the notation in Chapter 2 and omit the ⋅ when taking the scalar product for two vector valued functions.As a direct consequence of Proposition 2.1, we obtain the following variance bound for linear functionals. . BRASCAMP-LIEB INEQUALITY 29 Corollary . Let β be sufficiently large. For every f ∶ ◻ → R ( d ) , there exists C = C ( d, β ) < ∞ such that (2.2) var µ β, ◻ [( f, φ )] ≤ C ∑ x,y ∈◻ ○ ∣ f ( x )∣ ∣ G C (◻) ( x, y )∣ ∣ f ( y )∣ . Moreover, for any t ∈ R , ⟨ exp [ t ( f, φ )]⟩ µ β, ◻ ≤ exp ⎛⎝ Ct ∑ x,y ∈◻ ○ ∣ f ( x )∣ ∣ G C (◻) ( x, y )∣ ∣ f ( y )∣⎞⎠ . Proof.
The variance bound is a direct consequence of (2.1). To prove the bound for exponential moments,we differentiate the quantity(2.3) ∂ ∂t log ⟨ exp ( t ( f, φ ))⟩ µ β, ◻ = var µ t [( f, φ )] , where µ t satisfies dµ t dµ β, ◻ = Const × exp ( t ( f, φ )) . In other words, µ t is obtained from µ β, ◻ by adding a linear tilt, and is therefore log concave. The varianceestimate (2.2) for measure µ t in place of µ can be obtained without any changes to the arguments. The claimthus follows from integrating (2.2) for µ t . (cid:3) Corollary . Let β be sufficiently large. For every F ∶ R → R , F ′′ ∈ L ∞ ( R ) and ∣ g ( β, q )∣ ≤ exp (− β / ∥ q ∥ ) , we have ⟨ exp ⎛⎝ ∑ q ∈Q ◻ g ( β, q ) F ( φ, q )⎞⎠⟩ µ β, ◻ ≤ exp ⎛⎝ µ GFF, ◻ ⎛⎝ ∑ q ∈Q ◻ g ( β, q ) F ( φ, q )⎞⎠⎞⎠ , where dµ GF F, ◻ = Z GF F, ◻ × exp (− β ( φ, ∆ φ )) φ ∈ C (◻) dφ. Proof.
Similar to (2.3), we have(2.4) ∂ ∂t log ⟨ exp ⎛⎝ ∑ q ∈Q ◻ g ( β, q ) F ( φ, q )⎞⎠⟩ µ β, ◻ = var µ t ⎛⎝ ∑ q ∈Q ◻ g ( β, q ) F ( φ, q )⎞⎠ , where µ t is defined via density dµ t dµ β, ◻ = Const × exp ⎛⎝ t ∑ q ∈Q ◻ g ( β, q ) F ( φ, q )⎞⎠ . Denote by H β, ◻ and H t the Hamiltonian associated with the Gibbs measures µ β, ◻ and µ t . We then have forall x, y ∈ ◻ , ∂ x ⊗ ∂ y H t = ∂ x ⊗ ∂ y H β, ◻ + ∑ q ∈Q ◻ ,x,y z ( β, q ) F ′′ ( φ, q ) q ( x ) ⊗ q ( y ) , where ⊗ denotes the tensor product of two vectors (see (1.5) of Chapter 2). Using the estimate ∣ z ( β, q )∣ ≤ exp (− β / ∥ q ∥ ) , we see that for β sufficiently large, RRRRRRRRRRRR ∑ q ∈Q ◻ ,x,y z ( β, q ) F ′′ ( φ, q ) q ( x ) ⊗ q ( y )RRRRRRRRRRRR ≤ C e − c β / . Therefore for t ∈ [ , ] , ∂ x ⊗ ∂ y H t ≥ β − ∆. The claim thus follows from integrating t . (cid:3)
3. Coupling the finite volume Gibbs measures
The finite-volume Gibbs measures µ β,L can be realized as the invariant measure of a Markov process,known as the Langevin dynamics. Consider the diffusion process { φ t } ∶ ◻ × R → R ( d ) evolving according to(3.1) ⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩ dφ t = ⎡⎢⎢⎢⎢⎣ β ∆ φ t − ∑ n ≥ β β n / (− ∆ ) n + φ t − ∑ q ∈Q πz ( β, q ) q sin 2 π ( q, φ t )⎤⎥⎥⎥⎥⎦ dt + √ dB t ,φ t ∈ C (◻) , where ( B t ) t ≥ is a brownian motion on the space C (◻) (equipped with the standard L scalar product). Theinfinitesimal generator of this process is the operator − ∆ φ defined by(3.2) − ∆ φ F ( φ ) ∶= ∑ x ∈◻ ○ ∂ x F ( φ )− ∑ x ∈◻ ○ ⎡⎢⎢⎢⎢⎣ β ∆ φ ( x ) − ∑ n ≥ β β n / (− ∆ ) n + φ ( x ) − ∑ q ∈Q πz ( β, q ) q ( x ) sin 2 π ( q, φ )⎤⎥⎥⎥⎥⎦ ∂ x F ( φ ) , where the notation ∂ x means ∑ ( d ) i = ∂ x,i and we implicitly take the scalar product between the two terms inthe right side of (3.2). The domain of the operator ∆ φ includes the space of twice differentiable compactlysupported functions on the set C (◻) denoted by C c ( C (◻)) . Notice that we can write ∆ φ as∆ φ F = ∑ x ∈◻ ○ ∂ ∗ x ⋅ ∂ x F, where ∂ ∗ x denotes the formal adjoint of ∂ x with respect to µ β,L , given by ∂ ∗ x w ∶= − ∂ x w + ⎡⎢⎢⎢⎢⎣ β ∆ φ ( x ) − ∑ n ≥ β β n / (− ∆ ) n + φ ( x ) − ∑ q ∈Q πz ( β, q ) q ( x ) sin 2 π ( q, φ )⎤⎥⎥⎥⎥⎦ w ( φ ) . The operator ∆ φ is thus symmetric with respect to the measure µ β,L , and we define the Dirichlet form E (
F, G ) ∶= ⟨ F ∆ φ G ⟩ µ β,L = ⟨ G ∆ φ F ⟩ µ β,L = ∑ x ∈ Z d ⟨ ∂ x F, ∂ x G ⟩ µ β,L , ∀ F, G ∈ C c ( C (◻)) . In particular,(3.3) ∣⟨ G ∆ φ F ⟩ µ β,L ∣ ≤ ∥ F ∥ H ( µ β,L ) ∥ G ∥ H ( µ β,L ) , ∀ F, G ∈ C c ( C (◻)) , where we define the norm ∥ ⋅ ∥ H ( µ β,L ) by ∥ F ∥ H ( µ β,L ) ∶= ⟨∣ F ∣ ⟩ µ β,L + ( ∑ x ∈◻ ○ ⟨∣ ∂ x F ∣ ⟩ µ β,L ) . Let H ( µ β,L ) be the completion of C c ( C (◻)) with respect to the norm ∥ ⋅ ∥ H ( µ β,L ) . By (3.3) and a densityargument, the domain of the Dirichlet form E can be extended so that it includes the space H ( µ β,L ) , and wehave(3.4) ⟨ G ∆ φ F ⟩ µ β,L = ∑ x ∈◻ ⟨ ∂ x F, ∂ x G ⟩ µ β,L , ∀ F, G ∈ H ( µ β,L ) . The Langevin dynamics provides a convenient way to construct couplingsbetween different finite volume Gibbs measures. In the context of the gradient Gibbs measures with uniformlyconvex potential, this coupling technique was first used by Funaki and Spohn to prove the uniqueness of theinfinite volume Gibbs state [ ], and later used by [ ] to prove the CLT in finite domains and by [ ] to obtainquantitative rate of convergence for the Hessian of finite volume surface tensions. We will use this technique toobtain estimates on the difference of the ∇ φ fields corresponding to different underlying Gibbs measures µ β,L and µ β,M for different M, L ∈ N .The basic idea is that we can couple the measures by driving the dynamics in (3.1) with the samefamily { B t ( x )} of Brownian motions and estimating the difference of the solutions of the system of SDEswith the aid of parabolic estimates. Specifically, we will apply the C , − ε -regularity estimate for solutions ofparabolic equations with small ellipticity contrast, proved in (3.11) of Chapter 5.We denote by P ′ L,φ and P ′ M, ̃ φ the laws of the solution to (3.1) in the cubes ◻ L and ◻ M , starting from theinitial data φ and ̃ φ respectively. We use the symbol ⊗ to denote the product of measures. . COUPLING THE FINITE VOLUME GIBBS MEASURES 31 Proposition µ β,L and µ β,M ) . There exist constants β ∶= β ( d ) such that thefollowing statement holds for all β > β . Let L, M ∈ N with ≤ L ≤ M ≤ e L and let the finite volume measures µ β,L and µ β,M be defined as above. There exists a random element (∇ φ, ∇̃ φ ) of C ( R + ; Ω (◻ L ))× C ( R + ; Ω (◻ M )) with law Θ such that: (3.5) the law of ∇ φ is µ β,L ⊗ P ′ L,φ , (3.6) the law of ∇̃ φ is µ β,M ⊗ P ′ M, ̃ φ ,and a constant ε ∈ ( , ] , such that for all L > , there exists C = C ( d ) < ∞ such that (3.7) E Θ [ sup x ∈◻ L / ∣∇ φ ( x ) − ∇̃ φ ( x )∣] ≤ CL − + ε . Proof.
Let P ′ L,φ and P ′ M, ̃ φ be law of the Langevin dynamics (3.1) in the cubes ◻ L and ◻ M , startingfrom the initial data φ and ̃ φ respectively. We may couple these measures by requiring that the family { B t ( x ) ∶ x ∈ ◻ ○ L } of Brownian motions driving the dynamics are the same.We let P ∗( φ , ̃ φ ) be the resulting coupled measure of the joint process ( φ, ̃ φ ) . In other words, P ∗( φ , ̃ φ ) is thelaw on the set of trajectories ( φ t , ̃ φ t ) satisfying the coupled set of equations(3.8) ⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩ dφ t = ⎡⎢⎢⎢⎢⎣ β ∆ φ t − ∑ n ≥ β β n / (− ∆ ) n + φ t − ∑ q ∈Q ◻ L πz ( β, q ) q sin 2 π ( q, φ t )⎤⎥⎥⎥⎥⎦ dt + √ dB t ,d ̃ φ t = ⎡⎢⎢⎢⎢⎣ β ∆ ̃ φ t − ∑ n ≥ β β n / (− ∆ ) n + ̃ φ t − ∑ q ∈Q ◻ M πz ( β, q ) q sin 2 π ( q, ̃ φ t )⎤⎥⎥⎥⎥⎦ dt + √ dB t ,φ t ∈ C (◻ L ) , ̃ φ t ∈ C (◻ M ) , with initial data ( φ , ̃ φ ) . Let us sample the initial data with µ β,L × µ β,M itself by setting(3.9) Θ ′ ∶= ( µ β,L × µ β,M ) ⊗ P ∗( φ, ̃ φ ) . In other words, Θ ′ is the law of the pair ( φ t , ̃ φ t ) of trajectories obtained by first sampling φ and ̃ φ accordingto the measures µ β,L and µ β,M , respectively, and then running the dynamics (3.8).It is clear, by the invariance of the Gibbs measures with respect to the dynamics, that at any time t , thelaw of φ t is µ β,L and the law of ̃ φ t is µ β,M . We will eventually take the measure Θ as in the statement of theproposition to be the law of ( φ t , ̃ φ t ) at a given time t ∗ which will be selected below. This ensures that (3.5)and (3.6) are satisfied. It remains therefore to show that we can select t ∗ in such a way that the bound (3.7)is satisfied.Consider the difference(3.10) u ( t, x ) ∶= φ t ( x ) − ̃ φ t ( x ) , ( t, x ) ∈ ( , ∞) × ◻ L . Observe that u satisfies the parabolic equation, for any ( t, x ) ∈ ( , ∞) × ◻ L ,(3.11) ∂ t u ( t, x ) − β ∆ u ( t, x ) − ∑ n ≥ β β n / (− ∆ ) n + u ( t, x )− ∑ q ∈Q ◻ L ∇ ∗ q ⋅ ˆ a q ∇ q u ( t, ⋅) − ∑ q ∈Q ◻ M ,x ∖Q ◻ L πz ( β, q ) q ( x ) sin 2 π ( q, ̃ φ t ) = , where ∇ ∗ q ⋅ ˆ a q ∇ q is defined by(3.12) ∇ ∗ q ⋅ ˆ a q ∇ q u ∶= ( π ) z ( β, q ) q ( x )( u, q ) ∫ cos ( πs ( q, φ t ) + π ( − s )( q, ̃ φ t )) ds, Denote, for r > ( t, x ) ∈ ( , ∞) × Z d , the parabolic cylinder Q r ( t, x ) ∶= ( t, x ) + (− r , ] × ◻ r . We first notice that the term involving q ∈ Q ◻ M ,x ∖ Q ◻ L is exponentially small in L . Indeed, since the numberof lattice animal with diameter r grows exponentially in r , using the estimate ∣ z ( β, q )∣ ≤ e − β / ∥ q ∥ , we maytake β > β ( d ) such that RRRRRRRRRRRR ∑ q ∈Q ◻ M ,x ∖Q ◻ L πz ( β, q ) q ( x ) sin 2 π ( q, ̃ φ t )RRRRRRRRRRRR ≤ C M ∑ r = L e − r ≤ e − cL . Using the estimate ∣ z ( β, q )∣ ≤ e − cβ / ∥ q ∥ , we see that for large β , (3.11) is a small perturbation of the heatequation ∂ t u + β ∆ u =
0, and therefore the solution possess very strong regularity. By the C , − ε regularityestimate for the parabolic equation (3.11), which can deduced from the arguments presented Proposition 2.4and (3.12) of Chapter 5 with a minor adaptation to include the case of nonzero but small right-hand side, weobtain that if β is chosen large enough, then there exist ε ∈ ( , ] and C < ∞ such that for every t ∈ [ L , ∞) ,(3.13) L − ε [ u ] C , − ε ( Q L ( t, )) ≤ C ∥ u − ( u ) Q L ( t, ) ∥ L ( Q L ( t, )) + ∫ t − L + t ∑ y ∈ Z d e − c ( ln β )( L ∨∣ y ∣) ∣ u ( s, y )∣ ds + e − cL We claim that the second term on the right side acts as a small perturbation when β is large. To see this,observe that for β large enough, ∫ t − L + t ∑ y ∈◻ L e − c ( ln β )( L ∨∣ y ∣) ∣ u ( s, y )∣ ds ≤ e − cL ∥ u ∥ L ( Q L ( t, )) In particular, since we are on discrete lattice, one may divide by L − ε and apply the triangular inequality toobtain 12 sup x ∈◻ ⌈ L / ⌉ ∣∇ φ ( x ) − ∇̃ φ ( x )∣ ≤ [ u ] C , − ε ( Q ⌈ L / ⌉ ( t, )) (3.14) ≤ CL − + ε (∥ φ ∥ L ( Q L ( t, )) + ∥̃ φ ∥ L ( Q L ( t, )) +) + CL − + ε ∫ t − L + t ∑ y ∈◻ M ∖◻ L e − c ( ln β )∣ y ∣ ∣̃ φ ( s, y )∣ ds + e − cL Applying Lemma 3.3 below, we obtain for some C < ∞ , all L >
4, all t > s > P Θ [∥ φ ∥ L ( Q L ( t, )) + ∥̃ φ ∥ L ( Q L ( t, )) > Cs √ log Lt ] ≤ exp (− cs log ( Lt )) . We then use the exponential smallness of e − c ( ln β ) L to absorb any polynomial in L , and obtain P Θ ⎡⎢⎢⎢⎣RRRRRRRRRRR∫ t − L + t ∑ y ∈◻ M ∖◻ L e − c ( ln β )∣ y ∣ ∣̃ φ ( s, y )∣ ds RRRRRRRRRRR > Cs √ log M √ L ⎤⎥⎥⎥⎦≤ P Θ ⎡⎢⎢⎢⎢⎣ e − c ( ln β ) L ∣ ◻ M ∣ max ( t,x )∈[ , L ]×◻ M ∣̃ φ ( t, x )∣ > Cs √ log M √ L . ⎤⎥⎥⎥⎥⎦
Since M < e L , we may choose β ( d ) large enough so that e − c ( ln β ) L ∣ ◻ M ∣ < √ L , apply Lemma 3.3 again to ∣ ◻ M ∣ we see that the above probability is bounded by exp (− cs log ( M t )) . Since √ log M √ L <
1, taking t = L and takeexpectation with respect to Θ in (3.14), we conclude the proposition. (cid:3) The following lemma is a direct consequence of the Brascamp-Lieb inequality.
Lemma . Let β be sufficiently large. There exists C ( d, β ) < ∞ , such that, for every s ≥ C and L ∈ N , (3.15) µ β,L ( max x ∈◻ L ∣ φ ( x )∣ > Cs √ log L ) ≤ exp (− s log L ) . Proof.
We will prove (3.15) by estimating the exponential moments of ∣ φ ( x )∣ for each x ∈ ◻ L , , which wedo by an application of the Brascamp-Lieb inequality (Corollary 2.2), and then take a union bound over x .We obtain, for a constant C ( d, β ) < ∞ , and all s ∈ R ,max x ∈ Q L ⟨ exp ( s ∣ φ ( x )∣)⟩ µ β,L ≤ exp ( cs max x ∈ Q L ( G ◻ L ( x, x ))) ≤ exp ( Cs ) . . COUPLING THE FINITE VOLUME GIBBS MEASURES 33 Applying the Chebyshev inequality and optimize over s , we obtain, for a constant C ( d, β ) < ∞ and every s >
0, max x ∈ Q L µ β,L {∣ φ ( x )∣ > C s √ log L } ≤ exp (− s log L ) . The claim follows by taking a union bound over all x . (cid:3) We next give an estimate on the oscillations of the dynamical field ∣ φ t ∣ . Lemma . Let β be sufficiently large. There exists C ( d, β ) < ∞ such that, for every T, s ∈ ( , ∞) and L > , (3.16) ( µ β,L ⊗ P ′ L,φ ) [ max ( t,x )∈( ,T ]×◻ L ∣ φ t ( x )∣ > Cs √ log ( LT )] ≤ exp (− s ( log ( LT ))) . Proof.
Since the time parameter is continuous, we prove the claim in two steps. First we discretize the timeinto intervals of length ( log L ) − , and define the corresponding comb set by C ∶= {( t, x ) ∈ ( , T ] × ◻ L , t log L ∈ Z } .A union bound over the tail estimate proved in Lemma 3.2 controls the maximum of ∣ φ t ∣ over ( t, x ) ∈ C . Thenwe use continuity of the Brownian motion to bound ∣ φ t ( x ) − φ t ( x )∣ , whenever ∣ t − t ∣ < ( log L ) − .We first discuss the continuity estimates in t . The dynamics (3.1) imply, for every e = ( x, y ) ∈ E (◻ L ) , d ∇ φ t ( e ) = ⎡⎢⎢⎢⎢⎣ β ∆ ∇ φ t ( e ) − ∑ n ≥ β β n / (− ∆ ) n + ∇ φ t ( e ) − ⎛⎝ ∑ q ∈Q ◻ ,x − ∑ q ∈Q ◻ ,y ⎞⎠ πz ( β, q ) q ( x ) sin 2 π ( q, φ t )⎤⎥⎥⎥⎥⎦ dt + dB t ( e ) , where B t ( e ) ∶= √ ( B t ( y ) − B t ( x )) is a standard R ( d ) -valued Brownian motion. Let G t ∶= max e ∈ E (◻ L ) ∣∇ φ t ( e )∣ and M ∶= max e ∈ E (◻ L ) max t ∈( , ( log L ) − ] ∣ B t ( e )∣ , we use the fact that ∆ is a bounded operator on Z d , ∑ q ( x ) = ( q, φ t ) is a linear combination of ∇ φ t ) and the estimate z ( β, q ) ≤ exp (− β / ∥ q ∥ ) to conclude that forall large β , there exists C = C ( d, β ) so that G t ≤ C ∫ t G s ds + M. We apply Gronwall inequality to obtain for t ∈ ( , ( log L ) − ] G t ≤ ( M + ) + C ∫ t ( M + ) exp ( C ( t − s )) ds. That is, G t ≤ C ( M + ) exp ( C t ) . We now bound ∣ φ t ∣ by a comparison with independent Brownian motions. Denote by Ψ t ∶= φ t − (√ B t + φ ) .We then have for all x ∈ ◻ L , there exists C = C ( d, β ) such that ∣ d Ψ t ( x ) dt ∣ ≤ C ( M + ) exp ( C t ) . Integrating over t ∈ ( , ( log L ) − ] , we have the following inequality in law:max t ∈( , ( log L ) − ] ∣ Ψ t ( x )∣ ≤ C ( M + ) . We are now ready to finish the proof of the lemma. Given t ∈ ( , T ] , take t ∗ ∈ L Z such that t − t ∗ ∈ ( , ( log L ) − ] . Using the stationarity of φ t in time, we have the following inequalities in law:max ( t,x )∈( ,T ]×◻ L ∣ φ t ( x )∣ (3.17) ≤ max ( t,x )∈C ∣ φ t ( x )∣ + max ( t,x )∈( ,T ]×◻ L ∣ φ t ( x ) − φ t ∗ ( x )∣≤ max ( t,x )∈C ∣ φ t ( x )∣ + max ( t ∗ ,x )∈C max t ∈( , ( log L ) − ] ∣ φ t + t ∗ ( x ) − φ t ∗ ( x )∣≤ max ( t,x )∈C ∣ φ t ( x )∣ + max ( t ∗ ,x )∈C max t ∈( , ( log L ) − ] ∣ Ψ t ( x )∣ + ( t ∗ ,x )∈C max t ∈( , ( log L ) − ] ∣ B t ( x )∣≤ max ( t,x )∈C ∣ φ t ( x )∣ + C M + ( t ∗ ,x )∈C max t ∈( , ( log L ) − ] ∣ B t ( x )∣ . Applying Lemma 3.2 and taking a union bound over t ∈ ( log L ) − Z , we find, for L > ( µ β,L ⊗ P ′ L,φ ) [ max ( t,x )∈C ∣ φ t ( x )∣ > Cs √ log ( LT )] ≤ T log L exp (− s log ( LT )) ≤ exp (− s ( LT )) . Applying a union bound and then Doob’s inequality, we obtain ( µ β,L ⊗ P ′ L,φ ) [ M > s log ( LT )]≤ ∣ Q L ∣ ( µ L,ξ, per ⊗ P ′ L,ξ, per ,φ ) [ max t ∈( , ( log L ) − ] ∣ B t ( )∣ ≥ log L ]≤ ∣ Q L ∣ exp (− ( log L ) ) ≤ exp (− ( log L ) ) . Taking a union bound over ( t ∗ , x ) ∈ C then yields ( µ β,L ⊗ P ′ L,φ ) [ max ( t ∗ ,x )∈C max t ∈( , ( log L ) − ] ∣ B t ( x )∣ > Cs log ( LT )] ≤ exp (− ( log L ) ) . Combining (3.17) with the last three inequalities we conclude the lemma. (cid:3)
The coupling result from the previous subsection allows us to prove thethermodynamic limit of the measure µ β,L as L → ∞ . For β < ∞ , define the infinite volume Gibbs measureformally by(3.18) dµ β ( φ ) = Const × exp ⎛⎝ β ( φ, ∆ φ ) − ∑ n ≥ β β n / ( φ, (− ∆ ) n + φ ) + ∑ q ∈Q z ( β, q ) cos 2 π ( q, φ )⎞⎠ . Since µ β is a translation-invariant, ergodic Gibbs measure that only depends on ∇ φ , we uniquely determinethe Gibbs state by requiring that ⟨ φ ( )⟩ µ β = F ∈ C ∞ c ( Ω ) , the sequence of random variable ⟨ F (∇ φ )⟩ µ β,L , L ∈ N , forms a Cauchy sequence, thus converges as L → ∞ . Since ⟨ φ ( )⟩ µ β,L = β , it follows that thelaw of ∇ φ under µ β,L , viewed as an element of Ω, converges weakly as L → ∞ to µ β .Sending ∣ ◻ ∣ → ∞ in Corollary 2.2, and notice that all constants in the statement do not depend on thevolume, we obtain the Brascamp-Lieb inequality for the infinite volume measure µ β . Let G be the simplerandom walk Green’s function in Z d . Corollary . Let β be sufficiently large. For every f ∶ Z d → R ( d ) , there exists C = C ( d, β ) < ∞ suchthat (3.19) var µ β [( f, φ )] ≤ C ∑ x,y ∈ Z d ∣ f ( x )∣ G ( x, y )∣ f ( y )∣ . Moreover, for any t ∈ R , ⟨ exp [ t ( f, φ )]⟩ µ β ≤ exp ⎛⎝ Ct ∑ x,y ∈ Z d ∣ f ( x )∣ G ( x, y )∣ f ( y )∣⎞⎠ . An application of the Brascamp-Lieb inequality implies that for β sufficiently large, var µ β [∣ φ ( )∣] ≤ c β ∆ − ( , ) < ∞ , thus µ β is a well-defined φ -Gibbs measure. Combining with the thermodynamic limit resultsfor the Villain model [
18, 44 ], we are now ready to state the following dual representation in infinite volume.
Proposition . Let G be the simple random walk Green’s function in Z d . For β sufficiently large, wehave (3.20) ⟨ e i ( θ ( x )− θ ( )) ⟩ µ Vβ exp ( β G ( , x ))= ⟨ exp ⎛⎝ ∑ q ∈Q z ( β, q ) sin 2 π ( φ, q ) sin 2 π ( σ x , q ) + ∑ q ∈Q z ( β, q ) cos 2 π ( φ, q ) ( cos 2 π ( σ x , q ) − )⎞⎠⟩ µ β . COUPLING THE FINITE VOLUME GIBBS MEASURES 35 and (3.21) ⟨ e i ( θ ( x )+ θ ( )) ⟩ µ Vβ exp ( β G ( , x ))= ⟨ exp ⎛⎝ ∑ q ∈Q z ( β, q ) sin 2 π ( φ, q ) sin 2 π ( σ x , q ) + ∑ q ∈Q z ( β, q ) cos 2 π ( φ, q ) ( cos 2 π ( σ x , q ) − )⎞⎠⟩ µ β . Proof.
We give the proof of (3.20) below, (3.21) follows from the same argument. The thermodynamiclimit of the Villain model implies ⟨ e i ( θ ( x )− θ ( )) ⟩ µ Vβ,L → ⟨ e i ( θ ( x )− θ ( )) ⟩ µ Vβ as L → ∞ [
18, 44 ], and we also have G ◻ L ( , x ) → G ( , x ) . Apply Proposition 1.1, it suffices to show the weak convergence of µ β,L to µ β impliesthe right side of (1.13) converges to that of (3.20).To simplify the notation, denote by X ∶= ∑ q ∈Q z ( β, q ) sin 2 π ( φ, q ) sin 2 π ( σ x , q ) + ∑ q ∈Q z ( β, q ) cos 2 π ( φ, q ) ( cos 2 π ( σ x , q ) − ) . and for any R < ∞ , define the truncation of X in ◻ R by X R ∶= ∑ q ∈Q ◻ R z ( β, q ) sin 2 π ( φ, q ) sin 2 π ( σ x , q ) + ∑ q ∈Q ◻ R z ( β, q ) cos 2 π ( φ, q ) ( cos 2 π ( σ x , q ) − ) . We then have for all ∣ x ∣ ≪ R ≪ L , ⟨ exp ( X L )⟩ µ β,L = ⟨ exp ( X R )⟩ µ β,L + ⟨ exp ( X R ) ( exp ( X L − X R ) − )⟩ µ β,L and ⟨ exp ( X )⟩ µ β = ⟨ exp ( X R )⟩ µ β + ⟨ exp ( X R ) ( exp ( X L − X R ) − )⟩ µ β . In particular, by applying Proposition 3.1 to the geometric scales 2 k L, k ∈ N , using the fact that ( φ, q ) is alinear functional of ∇ φ and the estimate ∣ z ( β, q )∣ ≤ exp (− β / ∥ q ∥ ) , we see that as L → ∞ , ∣⟨ exp ( X R )⟩ µ β,L − ⟨ exp ( X R )⟩ µ β ∣ ≤ ∑ k ≥ ∣⟨ exp ( X R )⟩ µ β, kL − ⟨ exp ( X R )⟩ µ β, k + L ∣≤ ∑ k ≥ e CR d ( − ε ) k L − ε ≤ Ce CR d L ( − ε )/ , which tends to 0 as L tends to infinity. We apply the H¨older and Brascamp-Lieb inequalities (Corollary 2.2,2.3 and 3.4) to obtain ⟨ exp ( X R ) ( exp ( X L − X R ) − )⟩ µ β,L ≤ ⟨ exp ( X R )⟩ / µ β,L ⟨( exp ( X L − X R ) − ) ⟩ / µ β,L ≤ exp ( C var µ Gβ,L X R ) ⟨( exp ( X L − X R ) − ) ⟩ µ β,L . We claim that the Brascamp-Lieb inequality implies ⟨( exp ( X L − X R ) − ) ⟩ µ β,L ≤ C var µ Gβ,L ( X L − X R ) . Indeed, we can Taylor expand the left side and obtain
LHS = ⟨( ∑ k ≥ k ! ( X L − X R ) k ) ⟩ µ β,L = ∑ k ≥ ⟨ k ∑ j = j ! ( k − j ) ! ( X L − X R ) k ⟩ µ β,L . We may then apply the exponential Brascamp-Lieb inequality to even moments of X L − X R and the WickTheorem for the Gaussian measures to conclude ⟨( X L − X R ) k ⟩ µ β,L ≤ C k ⟨( X L − X R ) k ⟩ µ Gβ,L ≤ C k ( k ) ! k !2 k ( var µ Gβ,L ( X L − X R )) k . Therefore ⟨( exp ( X L − X R ) − ) ⟩ µ β,L ≤ ∑ k ≥ k + k ! C k ( var µ Gβ,L ( X L − X R )) k ≤ exp ( C var µ Gβ,L ( X L − X R )) − ≤ C var µ Gβ,L ( X L − X R ) . The same computation yields ⟨ exp ( X R ) ( exp ( X − X R ) − )⟩ µ β ≤ C exp ( C var µ Gβ X R ) var µ Gβ ( X − X R ) . We claim that for R ≫ ∣ x ∣ (e.g., R = e ∣ x ∣ ), both ⟨ exp ( X R ) ( exp ( X − X R ) − )⟩ µ β and ⟨ exp ( X R ) ( exp ( X L − X R ) − )⟩ µ β,L are bounded by R + ε − d for some ε ≪ µ Gβ,L X R < ∞ and var µ Gβ X R < ∞ . Also, for R ≫ ∣ x ∣ and r > R we have for every charge q ∈ Q such that supp q ∩ ◻ r ∖ ◻ r ≠ ∅ ,(3.22) ∣ z ( β, q ) sin 2 π ( σ x , q )∣ ≤ π exp (− β ∥ q ∥ )∣( σ x , q )∣ ≤ C ( r d − − ( r − ∣ x ∣) d − ) ≤ Cr d − − ε and similarly.(3.23) ∣ z ( β, q ) ( cos 2 π ( σ x , q ) − ) ∣ ≤ C exp (− β ∥ q ∥ )( σ x , q ) ≤ Cr d − − ε . To estimate var µ Gβ ( X − X R ) , we decompose X − X R = ∑ k ≥ ∆ X k R , where ∆ X k R ∶= ∑ q ∈Q ◻ k + R ∖Q ◻ kR z ( β, q ) sin 2 π ( φ, q ) sin 2 π ( σ x , q )+ ∑ q ∈Q ◻ k + R ∖Q ◻ kR z ( β, q ) cos 2 π ( φ, q ) ( cos 2 π ( σ x , q ) − ) . Using the estimates (3.22) and (3.23), we conclude thatvar µ Gβ ( ∆ X k R ) ≤ ( k R ) d − − ε . Therefore var µ Gβ ( X − X R ) ≤ ∑ k ∈ N var µ Gβ ∆ X k R ≤ CR d − − ε . Combining the estimates above we obtain ⟨ exp ( X R ) exp ( X − X R )⟩ µ β ≤ R d − − ε and similarly, ⟨ exp ( X R ) exp ( X L − X R )⟩ µ β,L ≤ R d − − ε . So that we conclude the proof. (cid:3)
Remark . By the same proof, one may obtain analogues of Proposition 3.1 for the Gibbs measures µ β, ◻ ,x ∗ and µ β, T ,x ∗ . One also observes that they converge to the same thermodynamic limit µ β . Therefore(3.20) also holds for the infinite volume measure µ Vβ, f , and the proofs in the rest of the paper applies to theVillain model with Neumann or periodic boundary conditions. . THE HELFFER-SJ ¨OSTRAND PDE 37
4. The Helffer-Sj¨ostrand PDE
Following the idea of [ ] (which was in turn inspired by the works [ ] and [ ]) and [ ], we will showin this section that the elliptic operator L , defined in (4.1) below, where∆ φ F ( φ ) ∶=∑ x ∈ F ( Z d ) ∂ x F ( φ ) − ∑ x ∈ Z d ⎡⎢⎢⎢⎢⎣ β ∆ φ ( x ) − ∑ n ≥ β β n / (− ∆ ) n + φ ( x ) − ∑ q ∈Q πz ( β, q ) q ( x ) sin 2 π ( q, φ )⎤⎥⎥⎥⎥⎦ ∂ x F ( φ ) , arises naturally when one considers the variance of certain observables with respect to the Gibbs measure µ β .In Section 4.1 we derive the Helffer-Sj¨ostrand representation, which identifies the variance of certain observ-ables under the measure µ β with the energy density of the Helffer-Sj¨ostrand PDE. In Section 4.2 we givevariational characterizations for the Helffer-Sj¨ostrand PDEs in the finite and infinite volume. The variationalcharacterization gives the solvability of the equation, and will be crucial to prove the main homogenizationresult for the Green’s function, which we state in Section 4.5. The proof here combines the ideas in [ ], [ ]and [ ], and extend them to the setting of differential forms and long range operators. The main result of this section is the Helffer-Sj¨ostrand rep-resentation for the Gibb measure µ β , stated as Proposition 4.1. To state the result, we introduce theHelffer-Sj¨ostrand operator(4.1) L ∶= ∆ φ − β ∆ + β ∑ n ≥ β n (− ∆ ) n + + ∑ q ∈Q ∇ ∗ q ⋅ a q ∇ q where we introduce the notation ∇ ∗ q ⋅ a q ∇ q u = π z ( β, q ) cos 2 π ( φ, q ) ( u, q ) q. Proposition . Fix
F, G ∈ H ( µ β ) and assume that there exist f, g ∶ Z d × Ω → R d × d which belong to thespace L ( Z d , µ β ) such that ∂ x F = ∇ ⋅ f ( x ) and ∂ x G = ∇ ⋅ g ( x ) . Then we have (4.2) cov µ β ( F ; G ) = ⟨( ∂F, L − ∂G )⟩ µ β = ⟨( f, ∇L − ∇ ⋅ g )⟩ µ β . Equivalently, we may write cov µ β ( F ; G ) = ∑ x ∈ Z d ⟨ ∂ x F, ∇ u ( x, ⋅)⟩ µ β , where u is the solution to the PDE L u = ∇ ⋅ g in Z d × Ω . Remark . The assumption that ∂F and ∂G are divergence of L ( Z d , µ β ) -vector fields is not essentialto prove the Helffer-Sj¨ostrand representation formula. Nevertheless, it simplifies the proof of this formula andis sufficient for the purposes of this article.By polarization, it suffices to prove the identity for variances in the above proposition. We would like toapply the integration by parts and obtain ⟨( F − ⟨ F ⟩ µ β ) ⟩ µ β = ⟨( F − ⟨ F ⟩ µ β ) ∆ φ ( ∆ φ ) − ( F − ⟨ F ⟩ µ β )⟩ µ β = ∑ x ∈ Z d ⟨( ∂ x F ) ( ∂ x ( ∆ − φ ( F − ⟨ F ⟩ µ β )))⟩ µ β . However, it is not clear that ∆ − φ ( F − ⟨ F ⟩ µ β ) is well-defined in the infinite volume. Therefore we will proveProposition 4.1 by first solving a PDE with a mass term λ and then send λ → φ F + λF = H, where ⟨ G ⟩ µ β = λ >
0. We have the unique variational solvability of (4.3) for any right-hand side G ∈ L ( µ β ) .This is stated in the next lemma. Lemma . Let G ∈ L ( µ β ) . Then there exists a solution F ∈ H ( µ β ) of the equation (4.4) ∆ φ F + λF = H. Moreover the solution F of (4.4) is unique, and there exists a constant C ( λ, β, d ) < ∞ such that ∥ F − ⟨ F ⟩ µ β ∥ H ( µ β ) ≤ C ∥ H ∥ L ( µ β ) . Proof.
This result can be obtained by an application of the Lax-Milgram lemma, or, alternatively, byconsidering the variational probleminf w ∈ H ( µ β ) ( ∑ x ∈ Z d ⟨( ∂ x w ) ⟩ µ β + λ ⟨ w ⟩ µ β − ⟨ Hw ⟩ µ β ) . In either case, we just use uniform coercivity with respect to the H ( µ β ) norm (and the coercivity dependson λ ). This problem can be solved equivalently by using tools of spectral theory as follows. The operator ∆ φ is well-defined on the space of smooth compactly supported functions defined on the space Ω and dependingon finitely many variables. Following the arguments of [
65, 40 ], we may extend this operator into a closed,self adjoint operator of L ( µ β ) which we still denote by ∆ φ . We denote by ( e − t ∆ φ ) t ≥ the L ( µ β ) -semigroupgenerated by this operator. Additionally, it follows from standard arguments (see [ ]), in view of the growthcondition imposed on the elements of Ω in Chapter 2, that for any initial condition φ ∈ Ω, the infinite volumeLangevin dynamics(4.5) ⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩ dφ t ( x ) = ⎡⎢⎢⎢⎢⎣ β ∆ φ t ( x ) − ∑ n ≥ β β n / (− ∆ ) n + φ t ( x ) − ∑ q ∈Q πz ( β, q ) q ( x ) sin 2 π ( q, φ t )⎤⎥⎥⎥⎥⎦ dt + √ dB t ( x ) , x ∈ Z d ,φ = φ is well-posed. We denote by ( P t ) t ≥ the probability transition semigroup associated to this dynamics. By thearguments of [ , Section 3 and Theorem 4.2] (as mentioned in [ ]), the two semigroup ( e − t ∆ φ ) t ≥ ( P t ) t ≥ coincide. In particular, we have the identities(4.6) w = ∫ ∞ e − λt e − t ∆ φ H d t = ∫ ∞ e − λt P t H d t. (cid:3) Applying the previous lemma and the integration by parts, we obtain for all λ > ⟨ F ⟩ µ β = ⟨ F ( ∆ φ + λ )( ∆ φ + λ ) − F ⟩ µ β = ∑ x ∈ Z d ⟨( ∂ x F ) ( ∂ x ( ∆ φ + λ ) − F )⟩ µ β + λ ⟨ F ( ∆ φ + λ ) − F ⟩ µ β . Using the fact that [ ∂ x , ∆ φ ] = ∑ y ∈ Z d ∂ x ∂ y H = β ∆ − β ∑ n ≥ β n (− ∆ ) n + − ∑ q ∈Q x ∇ ∗ q ⋅ a q ∇ q , we may rewrite the first term on the right side of (4.7) formally as ∑ x ∈ Z d ⟨( ∂ x F ) (L + λ ) − ( ∂F ) ( x, ⋅))⟩ µ β , where u = (L + λ ) − ( ∂F ) is the solution to the problem(4.8) ∆ φ u ( x, ⋅) + β ∆ u ( x, ⋅) − β ∑ n ≥ β n (− ∆ ) n + u ( x, ⋅) − ∑ q ∈Q x ∇ ∗ q ⋅ a q ∇ q u + λu ( x, ⋅) = ∂ x F (⋅) in Z d × Ω . We refer to [ ] for the rigorous justification of this step. In Lemma 4.4 below, we use a variational argumentto prove the solvability of the equation (4.8).By monotone convergence theorem, we see that the λ → ∑ x ∈ Z d ⟨( ∂ x F ) (L + λ ) − ∂ x F )⟩ µ β exists.We denote the limit by ∑ x ∈ Z d ⟨( ∂ x F )L − ( ∂F ) ( x, ⋅)⟩ µ β , and posit the existence of this object to Lemma 4.4.We next claim the second term in the right side of (4.7) converges to ⟨ F ⟩ µ β as λ →
0. Combining with theprevious argument implies(4.9) var µ β F = − ∑ x ∈ Z d ⟨( ∂ x F )L − ( ∂F ) ( x, ⋅)⟩ µ β , and the proof of Proposition 4.1 is complete. . THE HELFFER-SJ ¨OSTRAND PDE 39 By the identity (4.6) and the change of variable t → λt , we obtain λ ( ∆ φ + λ ) − F = ∫ ∞ λe −( ∆ φ + λ ) t F dt = ∫ ∞ λe − λt P t F dt = ∫ ∞ e − t P t / λ F dt.
It suffices to prove the right side converges in L ( µ β ) to ⟨ F ⟩ µ β . Indeed, ⟨(∫ ∞ e − t P t / λ F dt − ⟨ F ⟩ µ β ) ⟩ µ β = ⟨(∫ ∞ e − t ( P t / λ F − ⟨ F ⟩ µ β ) dt ) ⟩ µ β ≤ C ∫ ∞ e − t ⟨( P t / λ F − ⟨ F ⟩ µ β ) ⟩ µ β dt. Notice that for all s > ⟨( P s F − ⟨ F ⟩ µ β ) ⟩ µ β = ⟨( P s F ) ⟩ µ β − ⟨ F ⟩ µ β ≤ var µ β F , which follows from the fact that dds ⟨( P s F ) ⟩ µ β = ⟨( P s F ) dds ( P s F )⟩ µ β = − ⟨( P s F ) ∆ φ ( P s F )⟩ µ β ≤ . Using the ergodicity of the Langevin dynamics, we obtain the almost sure convergence P t F → ⟨ F ⟩ µ β as t → ∞ .We can thus apply the dominated convergence theorem and conclude that ⟨(∫ ∞ e − t P t / λ F dt − ⟨ F ⟩ µ β ) ⟩ µ β → In this section, we study the solvabilityof the Helffer-Sj¨ostrand equation in finite and infinite volume. The equations are solved variationally; thisapproach is one of the main techniques used in this article as it allows to prove quantitative homogenizationestimates on the solutions of the Helffer-Sj¨ostrand PDE (see Theorem 2 below).
The goal of this section is to prove that the equation − ∆ φ u + β ∆ u − β ∑ n ≥ β n (− ∆ ) n + u − ∑ q ∈Q x ∇ ∗ q ⋅ a q ∇ q u + λu = ∂ x F in Z d × Ω , for all λ ≥ ∂ x F = ∇ ⋅ f for some f ∈ L ( Z d , µ β ) .We now state the well-posedness for the Helffer-Sj¨ostrand equation in the infinite volume. Lemma . Assume that β is sufficiently large and select λ > . There exists a unique solution u λ ∈ H ( Z d , µ β ) of the equation (4.10) ∆ φ u λ − β ∆ u λ + β ∑ n ≥ β n (− ∆ ) n + u λ + ∑ q ∈Q ∇ ∗ q ⋅ a q ∇ q u λ + λu λ = ∇ ⋅ f in Z d × Ω , which satisfies, for a constant C ( d, β ) < ∞ , the estimate (4.11) λ ∥ u λ ∥ L ( Z d ,µ β ) + ∑ x ∈ Z d ∥ ∂ x u λ ∥ L ( Z d ,µ β ) + ∥∇ u λ ∥ L ( Z d ,µ β ) ≤ C ∥ f ∥ L ( Z d ,µ β ) . For each point x ∈ Z d , the map u λ converges weakly in the spaces L ( µ β ) as λ → . The weak limit u ∶= L∇ ⋅ g is the unique solution (up to a constant) of the equation ∆ φ u − β ∆ u + β ∑ n ≥ β n (− ∆ ) n + u + ∑ q ∈Q ∇ ∗ q ⋅ a q ∇ q u = −∇ ⋅ f in Z d × Ω , Proof.
The proof is variational. A function u λ ∈ H ( Z d , µ β ) is a solution of (4.10) if and only if(4.12) ∑ y ∈ Z d ∑ x ∈ Z d ⟨( ∂ y u λ ( x, ⋅))( ∂ y w ( x, ⋅))⟩ µ β + β ∑ x ∈ Z d ⟨∇ u λ ( x, ⋅)∇ w ( x, ⋅)⟩ µ β + β ∑ n ≥ β n ∑ x ∈ Z d ⟨∇ n + u λ ( x, ⋅) , ∇ n + w ( x, ⋅)⟩ µ β + ∑ x ∈ Z d λ ⟨ u λ ( x, ⋅) , w ( x, ⋅)⟩ µ β + ∑ q ∈Q ⟨∇ q u λ ⋅ a q ∇ q w ⟩ µ β = − ∑ x ∈ Z d ⟨ f ( x, ⋅)∇ w ( x, ⋅)⟩ µ β , ∀ w ∈ H ( Z d , µ β ) . Using the estimate ∣ a q ∣ ≤ Ce − β ∥ q ∥ , if β is sufficiently large, the symmetric bilinear form on the left side of theprevious display is coercive with respect to the H ( Z d , µ β ) norm. The Lax-Milgram lemma therefore yieldsthe existence of a unique solution u ∈ H ( Z d , µ β ) . We see that this function satisfies (4.11) by taking w = u λ in (4.12).By the Gagliardo-Nirenberg-Sobolev inequality, we see that for any λ > φ ∈ Ω, ∥ u λ ( x, ⋅)∥ L ( µ β ) ≤ ⟨( ∑ x ∈ Z d ∥ u λ ( x, φ )∥ dd − ) d − d ⟩ µ β ≤ ∑ x ∈ Z d ∥∇ u λ ∥ L ( µ β ) ≤ C. Combining the previous estimate with (4.11), we obtain that (up to extraction) for each point x ∈ Z d , the map u λ ( x, ⋅) converges weakly to a function u ( x, ⋅) ∈ L ( µ β ) and that (still up to extraction) the functions ∇ u λ and ∂u λ converge weakly to ∇ u and ∂u in the spaces L ( Z d , µ β ) and L ( Z d × Z d , µ β ) respectively. From (4.12),we deduce that u satisfies the identity: for each map w ∈ H ( Z d , µ β )∑ y ∈ Z d ∑ x ∈ Z d ⟨( ∂ y u ( x, ⋅))( ∂ y w ( x, ⋅))⟩ µ β + β ∑ x ∈ Z d ⟨∇ u ( x, ⋅)∇ w ( x, ⋅)⟩ µ β + β ∑ n ≥ β n ∑ x ∈ Z d ⟨∇ n + u ( x, ⋅) , ∇ n + w ( x, ⋅)⟩ µ β + ∑ q ∈Q ⟨∇ q u ⋅ a q ∇ q w ⟩ µ β = − ∑ x ∈ Z d ⟨ f ( x, ⋅)∇ w ( x, ⋅)⟩ µ β . (cid:3) We next present the well-posedness of the Dirichletboundary value problem for the Helffer-Sj¨ostrand equation in a cube ◻ ⊆ Z d . This will be used to prove theregularity of the solutions in Chapter 5 and to establish the convergence of subadditive quantities associatedwith the equations in Chapter 6. Lemma . Assume that β is sufficiently large. Let ◻ ⊆ Z d be a cube of size R , h ∈ L (◻ , µ β ) and u ∈ H (◻ , µ β ) . There exists a unique solution u ∈ H (◻ , µ β ) of the boundary-value problem (4.13) ⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩ − ∆ φ u + β ∆ u − β ∑ n ≥ β n (− ∆ ) n + u − ∑ q ∈Q , supp q ∩◻≠∅ ∇ ∗ q ⋅ a q ∇ q u = h in ◻ ○ × Ω ,u = u on ∂ ◻ × Ω , which satisfies, for a constant C ( d, β ) < ∞ , the estimate (4.14) ∥ u ∥ H (◻ ,µ β ) ≤ C ( R ∥ h ∥ L (◻ ,µ β ) + ∥ u ∥ H (◻ ,µ β ) ) . Proof.
A function u ∈ u + H (◻ , µ β ) is a solution of (4.13) if and only if(4.15) ∑ y ∈◻ ∑ x ∈◻ ⟨( ∂ y u ( x, ⋅))( ∂ y w ( x, ⋅))⟩ µ β + β ∑ x ∈◻ ⟨∇ u ( x, ⋅)∇ w ( x, ⋅)⟩ µ β + β ∑ n ≥ β n ∑ x ∈◻ ⟨∇ n + u ( x, ⋅) , ∇ n + w ( x, ⋅)⟩ µ β + ∑ q ∈Q , supp q ∩◻≠∅ ⟨∇ q u ⋅ a q ∇ q w ⟩ µ β = ∑ x ∈◻ ⟨ h ( x, ⋅) w ( x, ⋅)⟩ µ β , ∀ w ∈ H (◻ , µ β ) . For large β , the symmetric bilinear form on the left side of the previous display is is a small perturbation ofthe Laplacian term, and therefore (cid:74) u (cid:75) H (◻ ,µ β ) ≤ C ( R ∥ h ∥ L (◻ ,µ β ) + ∥ u ∥ H (◻ ,µ β ) ) . Apply Lemma 1.1 (i), we seethat the bilinear form is coercive with respect to the H (◻ , µ β ) norm. The Lax-Milgram lemma thereforeyields the existence of a unique solution u ∈ H (◻ , µ β ) . It also follows that the solution to (4.13) admits avariational characterization: u minimizes the energy ∑ y ∈ Z d ∥ ∂ y u ∥ L (◻ ,µ β ) + β ∥∇ u ∥ L (◻ ,µ β ) + ∑ n ≥ β n ∥∇ n + u ∥ L ( Z d ,µ β ) + ∑ q ∈Q , supp q ∩◻≠∅ ⟨∇ q u ⋅ a q ∇ q u ⟩ µ β − ∑ x ∈◻ ⟨ h ( x, ⋅)∇ u ( x, ⋅)⟩ µ β , among all the functions in the space u + H (◻ , µ β ) . (cid:3) . THE HELFFER-SJ ¨OSTRAND PDE 41 In this section, we present recordsome properties of the Green’s matrix associated to the elliptic Helffer-Sj¨ostrand operator L . We highlightthat since the operator L is an elliptic system, the fundamental solution is a matrix; this object is usedrepeatedly in the following chapters as it allows to decompose the solution of the Helffer-Sj¨ostrand equationstated in (4.10). We fix an exponent p ∈ [ , ∞] , and a function f ∈ L p ( µ β ) and define the elliptic Green’smatrix G f ∶ Z d × Ω × Z d ↦ R ( d )×( d ) by the formula(4.16) LG f ( x, φ ; y ) = f ( φ ) δ y ( x ) in Z d × Ω , such that ∥G f ( x, ⋅ ; y )∥ L p ( µ β ) tends to 0 as x tends to infinity. To be slightly more precise in the definition, wesee the Dirac δ y to be the diagonal matrix δ y ( x ) ∶= ( x = y ⋅ i = j ) ≤ i,j ≤( d ) . To solve the equation (4.16), we fix acolumn in the matrix f δ y , solve the system (4.16) (with this specific column) and obtain a function valued inthe space R ( d ) . We then perform the same operation on the ( d ) − ( d ) solutions obtainedthis way to define the matrix G f (⋅ , ⋅ ; y ) .In the case p =
2, we can solve (4.16) variationally, by applying the Gagliardo-Nirenberg-Sobolev inequality.The solvability in the general case relies on the Feynman-Kac formula and tools from spectral theory aspresented in the introduction of Chapter 5 (following the ideas of [ , Section 2.2.2]) Lemma . There exists β = β ( d ) , such that for all β > β , there exists C = C ( d, β ) , such that theGreen’s function G f defined from (4.16) satisfies ∥G f ∥ L ∗ ( Z d ,L ( µ β )) ≤ C ∥ f ∥ L ( µ β ) . Proof.
The proof follows from the same argument as Lemma 4.4. We test (4.16) using G f , and observingthat for large β , L is a small perturbation of β ∆, to obtain ∥∇G f ∥ L ( Z d ,µ β ) ≤ C ⟨G f ( y ) f (⋅)⟩ µ β ≤ C ∥G f ∥ L ∗ ( Z d ,L ( µ β )) ∥ f ∥ L ( µ β ) . Applying the Gagliardo-Nirenberg-Sobolev inequality, we have ∥G f ∥ L ∗ ( Z d ,L ( µ β )) ≤ C ( d ) ∥∇G f ∥ L ( Z d ,µ β ) . (cid:3) The following proposition quantifies the solvability lemma and prove asymptotic decays on the L p ( µ β ) -norm of the Green’s matrix, its gradient and its mixed derivative. The proof is based on regularity theory forthe Helffer-Sj¨ostrand operator and is one of the main subject of Chapter 5. Proposition . For any regularity exponent ε > , there exists an inverse temperature β ( d, ε ) < ∞ such that the following result holds. For any β > β , there exists a constant C ( d ) < ∞ such that for any x ∈ Z d ,one has the estimates ∥G f ( x, ⋅ ; y )∥ L p ( µ β ) ≤ Cβ ∥ f ∥ L p ( µ β ) ∣ x − y ∣ d − , and the regularity estimates on the gradient and the mixed derivative ∥∇ x G f ( x, ⋅ ; y )∥ L p ( µ β ) ≤ Cβ ∥ f ∥ L p ( µ β ) ∣ x − y ∣ d − − ε and ∥∇ x ∇ y G f ( x, ⋅ ; y )∥ L p ( µ β ) ≤ Cβ ∥ f ∥ L p ( µ β ) ∣ x − y ∣ d − ε . In this section, we state aquantitative result which establishes homogenization for the mixed gradient of the Green’s matrix associatedto the Helffer-Sj¨ostrand operator. The proof of the theorem below will be presented in Chapters 6 and 7.Naturally, one expects that the Green’s matrix associated to the Helffer-Sj¨ostrand operator (4.1), defined by L G = δ in Z d × Ωhomogenizes to the Green’s matrix G associated to the Laplacian operator ∇ ⋅ a β ∇ :(4.17) − ∇ ⋅ a β ∇ G = δ in Z d , where a β is a positive definite matrix which is a small perturbation of the matrix β I d . It is defined inChapters 6 and 7 as the limit of the energy associated to the Dirichlet problem with affine boundary condition(see Definition 1.5 and Corollary 2.3 in Chapter 6 and the introduction of Chapter 7); it is deterministic anddepends only on the dimension d and the inverse temperature β . The solvability of the equation (4.17) isensured by the fact that a β is a perturbation of a diagonal matrix and the arguments of Chapter 5.When applying to the Villain model (see computations in Chapter 4) we need a largely nonlineargeneralization of the homogenization result (4.17) and also the convergence of the mixed gradient of theGreen’s matrix. We present as Theorem 2 below and the proof is deferred to Chapters 6 and 7. For each pair of integers ( i, j ) ∈ { , . . . , d } × { , . . . , ( d )} , we let l e ij be the linear function defined by theformula l e ij ∶= ⎧⎪⎪⎨⎪⎪⎩ R d → R ( d ) ,x → ( , . . . , x ⋅ e i , . . . , ) , where the term x ⋅ e i appears in the j -th position. We denote by ∇ χ ij the gradient of the infinite-volumecorrector, which is the unique stationary solution of the Helffer-Sj¨ostrand equation L ( l e ij + χ ij ) = Z d × Ω . It is constructed as the infinite volume limit of finite volume correctors, the later measures the homogeniza-tion error for the Helffer-Sj¨ostrand equation with affine boundary condition. For a precise definition, seeProposition 4.4 of Chapter 6. Once equipped with the gradient of the corrector, we can define the exteriorderivative d ∗ χ ij by using that the codifferential d ∗ is a linear functional of the gradient (see (2.10)). Thefollowing theorem proves a quantitative homogenization result for a version of the mixed derivative of theGreen’s function (4.16), the specific form of the function (4.18) is justified by the fact that it is the correctobject to consider in order to prove Theorem 1 in Chapter 4. We mention that the techniques developedin Chapters 6 and 7 can be adapted to prove quantitative homogenization of more general solutions of theHelffer-Sj¨ostrand equation. Theorem . We fix a charge q ∈ Q suchthat belongs to the support of n q , let U q be the solution of the Helffer-Sj¨ostrand equation (4.18) LU q = cos 2 π ( φ, q ) q in Z d × Ω , and let G q ∶= ( G q , , . . . , G q , ( d ) ) be the map defined by the formula, for each integer k ∈ { , . . . , ( d )} , (4.19) G q ,k = ∑ ≤ i ≤ d ∑ ≤ j ≤( d ) ⟨ cos 2 π ( φ, q ) ( n q , d ∗ l e ij + d ∗ χ ij )⟩ µ β ∇ i G jk . There exist an inverse temperature β ∶= β ( d ) < ∞ , an exponent γ ∶= γ ( d ) > and a constant C q whichsatisfies the estimate ∣ C q ∣ ≤ C ∥ q ∥ k for some C ∶= C ( d, β ) < ∞ and k ∶= k ( d ) < ∞ , such that for each β ≥ β and each radius R ≥ , one has the inequality (4.20) XXXXXXXXXXXXX∇U q − ∑ ≤ i ≤ d ∑ ≤ j ≤( d ) ( e ij + ∇ χ ij ) ∇ i G q ,j XXXXXXXXXXXXX L ( B R ∖ B R ,µ β ) ≤ C q R d + γ . Remark . The functions ∇U q and ∇ i G q behaves like mixed derivative of Green’s matrices, inparticular, they should decay like the map x → ∣ x ∣ − d . Theorem 2 states that their difference is quantitativelycloser than the typical size of the two functions: we obtain an algebraic rate of convergence with additionalexponent γ > Remark . For the purposes of Chapter 4, we record here that the statement of Theorem 2 can besimplified by using the formalism of discrete differential forms. To this end, we recall the definition of theoperator L , d ∗ introduced in Section 2.2 of Chapter 2 record the following properties: ● The operator −∇ ⋅ a β ∇ can be written(4.21) − ∇ ⋅ a β ∇ = β ( d ∗ d + ( + λ β ) dd ∗ ) , where λ β is a real coefficient which is small tends to 0 as β tends to infinity. This property is statedin Remark 1.11 of Chapter 6; ● The gradient of the infinite volume corrector only depends on the value of the codifferential d ∗ l e ij (inparticular, it is equal to 0 if d ∗ l e ij =
0) as mentioned in Remarks 4.2 and 4.5 of Chapter 6. We use thenotation of Remark 4.5: given an integer k ∈ { , . . . , d } , we let select a vector p ∶= ∑ ≤ i ≤ d ∑ ≤ j ≤( d ) p ij e ij such that d l p = e k and denote by ∇ χ k ∶= ∑ ≤ i ≤ d ∑ ≤ j ≤( d ) p ij ∇ χ ij .Using these ingredients, we can rewrite the definition of the map G q ,k stated in (4.19): we have G q ,k = ∑ ≤ i ≤ d ⟨ cos 2 π ( φ, q ) ( n q , e i + d ∗ χ i )⟩ µ β ( d ∗ G ⋅ k ) ⋅ e i . . THE HELFFER-SJ ¨OSTRAND PDE 43 We then use that, by definition, the map G ⋅ ,k solves the equation −∇⋅ a β ∇ G = δ and the identities − ∆ = dd ∗ + d ∗ d,d ○ d = ∗ ○ d ∗ = − ( + λ β ) ∆d ∗ G = ( + λ β ) ( dd ∗ + d ∗ d ) d ∗ G ⋅ k = ( + λ β ) d ∗ dd ∗ G ⋅ k = d ∗ ( d ∗ d G ⋅ k + ( + λ β ) dd ∗ ) G ⋅ k = d ∗ (−∇ ⋅ a β ∇ G ⋅ ,k )= d ∗ δ . The exterior derivative d ∗ G can thus be explicitly computed in terms of the gradient of the Green’s functionassociated to the operator − ( + λ β ) ∆ which is equal to the the standard random walk Green’s function onthe lattice Z d multiplied by the value ( + λ β ) − .HAPTER 4 First-order expansion of the two-point function
In this chapter we show that by combining Theorem 2, which gives a quantitative rate of convergence ofthe mixed gradients of the Helffer-Sj¨ostrand Green’s matrix, with a regularity theory for the Helffer-Sj¨ostrandoperator, implies the convergence of the two-point function stated in Theorem 1. The proof relies on theregularity theory that is developed in Chapter 5.The objective of this chapter is to prove Theorem 1. To this end, by Proposition 3.5 of Chapter 3, it isenough to prove the expansion stated in the following theorem.
Theorem . There exist constants β ∶= β ( d ) , c ∶= c ( β, d ) , c ( β, d ) and an exponent γ ′ ∶= γ ′ ( d ) > such that for every β > β , and every x ∈ Z d , Z ( σ x ) Z ( ) = c + c ∣ x ∣ d − + O ( C ∣ x ∣ d − + γ ′ ) , and Z ( σ x ) Z ( ) = c + c ∣ x ∣ d − + O ( C ∣ x ∣ d − + γ ′ ) , The proof of Theorem 3 requires to use the following statements established in Chapter 5, Chapter 6 andChapter 7: ● We need to use the quantitative homogenization of the mixed derivative of the Green’s functionassociated to the Helffer-Sj¨ostrand equation L . The precise statement we need to use is given inTheorem 2. The proof of this theorem is the subject of Chapters 6 and 7; ● We need to use the C , − ε regularity theory established in Chapter 5; more specifically, we needto use the regularity estimates for the Green’s function G associated to the operator L stated inProposition 4.7 of Chapter 3 and on the Green’s function G der stated in Proposition 4.2 of Chapter 5.We additionally make the assumption that the regularity exponent ε is very small compared to theexponent γ which appears in the statement of Theorem 2 of Chapter 3 (for instance, we assumethat the ratio γε is larger than 100 d ). This condition can always be ensured by increasing the inversetemperature β .Apart from these three results, the proof of Theorem 3, which is contained in this chapter (and Chapter 8 forthe technical estimates) is largely independent from Chapters 5, 6 and 7.Finally, some computations presented in this chapter requires to prove estimates on terms of the form ∑ x ∈ Z d ∣ x ∣ α ∣ x − y ∣ β , for some exponents α, β > α + β > d . We refer to Appendix C for the proof of the upper boundsand directly write the results in the sections below.This chapter is organized as follows. We first set up the argument and introduce some preliminary notationsin Section 1. We then simplify the expression (1.1) below in a series of technical lemmas stated in Sections 2, 3and 4. In particular, in Sections 3 and 4, we sketch the argument that one can decouple the Helffer-Sj¨ostrandGreen’s matrix from the exponential terms arising from the dual model in Chapter 3. The proofs of theselemmas relies on the C , − ε -regularity theory established in Chapter 5, we give an outline of the arguments andpostpone the proof to Chapter 8. The core of the proof of Theorem 3 (thus Theorem 1) is given in Section 5.This section is decomposed into two subsections. We first write an outline of the argument in Section 1 andthen present the details of the proof in Section 5.2.
456 4. FIRST-ORDER EXPANSION OF THE TWO-POINT FUNCTION
1. Preliminary notations
We first recall that we have the identity(1.1) Z β ( σ x ) Z β ( ) = ⟨ exp ⎛⎝ ∑ q ∈Q z ( β, q ) sin 2 π ( φ, q ) sin 2 π ( σ x , q ) + ∑ q ∈Q z ( β, q ) cos 2 π ( φ, q ) ( cos 2 π ( σ x , q ) − )⎞⎠⟩ µ β . We also recall that, by the definition of the function σ x given in Section 1 of Chapter 3, we have the equalityd ∗ σ x = d ∗ d (− ∆ ) − h x = h ,x − dd ∗ (− ∆ ) − h x = h ,x − d (− ∆ ) − d ∗ h x = h ,x − d (− ∆ ) − ( x − )= h ,x + ∇ G − ∇ G x . We then use the identity q = d n q , that the maps q , n q and h ,x are valued in Z , and the periodicity of the sineand the cosine to deduce thatsin 2 π ( σ x , q ) = sin 2 π (∇ G − ∇ G x , n q ) and cos 2 π ( σ x , q ) = cos 2 π (∇ G − ∇ G x , n q ) . One can then expand the sine and the cosine by using the trigonometric formulas. We obtain the identities(1.2) sin 2 π (∇ G − ∇ G x , n q ) = sin 2 π (∇ G, n q ) − sin 2 π (∇ G x , n q )+ ( cos 2 π (∇ G x , n q ) − ) sin 2 π (∇ G, n q ) − ( cos 2 π (∇ G, n q ) − ) sin 2 π (∇ G x , n q ) , and(1.3) cos 2 π (∇ G − ∇ G x , n q ) − = ( cos 2 π (∇ G, n q ) − ) ( cos 2 π (∇ G x , n q ) − )+ ( cos 2 π (∇ G, n q ) − ) + ( cos 2 π (∇ G x , n q ) − ) + sin 2 π (∇ G, n q ) sin 2 π (∇ G x , n q ) . We then combine the identities (1.2) and (1.3) with the right side of (1.1). To ease the notation, we introducethe following random variables(1.4) ⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩ X x ∶= exp ⎛⎝− ∑ q ∈Q z ( β, q ) ( sin 2 π ( φ, q ) sin 2 π (∇ G x , n q ) −
12 cos 2 π ( φ, q ) ( cos 2 π (∇ G x , n q ) − ))⎞⎠ ,Y ∶= exp ⎛⎝ ∑ q ∈Q z ( β, q ) ( sin 2 π ( φ, q ) sin 2 π (∇ G, n q ) +
12 cos 2 π ( φ, q ) ( cos 2 π (∇ G, n q ) − ))⎞⎠ ,Y x ∶= exp ⎛⎝ ∑ q ∈Q z ( β, q ) ( sin 2 π ( φ, q ) sin 2 π (∇ G x , n q ) +
12 cos 2 π ( φ, q ) ( cos 2 π (∇ G x , n q ) − ))⎞⎠ ,X sin cos ∶= exp ⎛⎝− ∑ q ∈Q z ( β, q ) sin 2 π ( φ, q ) sin 2 π (∇ G x , n q ) ( cos 2 π (∇ G, n q ) − )⎞⎠× exp ⎛⎝ ∑ q ∈Q z ( β, q ) sin 2 π ( φ, q ) sin 2 π (∇ G, n q ) ( cos 2 π (∇ G x , n q ) − )⎞⎠ ,X cos cos ∶= exp ⎛⎝ ∑ q ∈Q z ( β, q ) cos 2 π ( φ, q ) ( cos 2 π (∇ G, n q ) − ) ( cos 2 π (∇ G x , n q ) − )⎞⎠ ,X sin sin ∶= exp ⎛⎝ ∑ q ∈Q z ( β, q ) cos 2 π ( φ, q ) sin 2 π (∇ G, n q ) sin 2 π (∇ G x , n q )⎞⎠ . In this notation we have(1.5) Z β ( σ x ) Z β ( ) = ⟨ Y X x X sin cos X cos cos X sin sin ⟩ µ β . Our aim is then to simplify the identity (1.5) and then to apply Theorem 4. . REMOVING THE CONTRIBUTIONS OF THE COSINES 47
2. Removing the terms X sin cos , X cos cos and X sin sin We first show that the terms X sin cos , X cos cos and X sin sin are lower order terms which can be removedfrom the analysis. We prove the following lemma. Lemma 1.1.
There exist constants β ∶= β ( d ) < ∞ , c ∶= c ( d, β ) and C ∶= C ( d, β ) such that for each β > β , (2.1) Z β ( σ x ) Z β ( ) = ⟨ Y X x ⟩ µ β + c ⟨ Y X x ⟩ µ β ∣ x ∣ d − + O ( C ∣ x ∣ d − ) . A consequence of the identity (2.1) is the equivalence ∃ c , c ∈ R , Z β ( σ x ) Z β ( ) = c + c ∣ x ∣ d − + O ( C ∣ x ∣ d − + γ ′ ) ⇐⇒ ∃ c , c ∈ R , ⟨ Y X x ⟩ µ β = c + c ∣ x ∣ d − + O ( C ∣ x ∣ d − + γ ′ ) . This lemma is technical and its proof is not the core of the argument; the proof is thus deferred toChapter 8. We provide here a sketch of the argument.
Sketch of the proof of Lemma 1.1.
To prove the identity (2.1), we first record four standard in-equalities, for each y ∈ Z d , and each a ∈ R ,(2.2) ∣∇ G ( y )∣ ≤ C ∣ y ∣ d − , ∣∇ G x ( y )∣ ≤ C ∣ y − x ∣ d − , ∣ sin a ∣ ≤ ∣ a ∣ and ∣ cos a − ∣ ≤ ∣ a ∣ . Using the estimates (2.2) and the exponential decay of the coefficient z ( β, q ) , we prove the following estimates:(i) The random variables X sin cos and X cos cos belongs to the space L ∞ ( µ β ) and satisfy the estimates(2.3) ⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩ ∥ X sin cos − ∥ L ∞ ≤ C ∣ x ∣ d − , ∥ X cos cos − ∥ L ∞ ≤ C ∣ x ∣ d − . (ii) We prove that the random variable X sin sin also belongs to the space L ∞ ( µ β ) and that its fluctuationsaround the value 1 are of order ∣ x ∣ − d . This is larger than the fluctuations of the random variables X sin cos and X cos cos and one needs to be more precise in the analysis: we prove the following estimateson the expectation and the variance X sin sin (2.4) ⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩ var µ β X sin sin ≤ C ∣ x ∣ d − , ⟨ X sin sin ⟩ µ β = + c ∣ x ∣ d − + O ( C ∣ x ∣ d − ) . The variance is estimated thanks to the Brascamp-Lieb inequality and the expectation is estimatedthanks to the estimates (2.2) and a Taylor expansion of the exponential.A combination of the estimates (2.3) and (2.4) is then sufficient to prove Lemma 1.1. (cid:3)
Remark . The same proof also yields(2.5) Z β ( σ x ) Z β ( ) = ⟨ Y Y x ⟩ µ β + c ⟨ Y Y x ⟩ µ β ∣ x ∣ d − + O ( C ∣ x ∣ d − ) . In general, c ≠ c since the O (∣ x ∣ d − ) term above is contributed by ⟨ X − ⟩ µ β instead of ⟨ X sin sin ⟩ µ β .
3. Removing the contributions of the cosines
From Lemma 1.1, we see that to prove Theorem 1, it is sufficient to obtain the following expansion(3.1) ∃ c , c ∈ R , ⟨ Y X x ⟩ µ β = c + c ∣ x ∣ d − + O ( C ∣ x ∣ d − + γ ′ ) . We then note that, by the translation invariance of the measure µ β , the expectation of the random variable X x does not depend on the point x : we have, for each x ∈ Z d , ⟨ X x ⟩ µ β = ⟨ X ⟩ µ β . A consequence of this observationis that to prove (3.1), it is sufficient to show(3.2) cov [ X x , Y ] = c ∣ x ∣ d − + O ( C ∣ x ∣ d − + γ ′ ) . Indeed, the expansion (3.2) implies (3.1) with the value c = ⟨ Y ⟩ µ β ⟨ X ⟩ µ β . To prove the identity (3.1), weuse the Helffer-Sj¨ostrand representation formula and write the covariance in the following form(3.3) cov [ X x , Y ] = ∑ y ∈ Z d ⟨( ∂ y X x ) Y( y, ⋅)⟩ µ β , where Y is the solution of the Helffer-Sj¨ostrand equation, for each ( y, φ ) ∈ Z d × Ω,(3.4)
LY( y, φ ) = ∂ y Y ( φ ) . For each point x ∈ Z d , we introduce the notation Q x to denote the (random) charge: for pair each ( y, φ ) ∈ Z d × Ω,(3.5) Q x ( y, φ ) ∶= π ∑ q ∈Q z ( β, q ) cos 2 π ( φ, q ) sin 2 π (∇ G x , n q ) q ( y ) . These charges are defined so as to have the identities, for each y ∈ Z d ,(3.6) ∂ y Y ( φ ) = ⎛⎝ Q ( y, φ ) −
12 2 π ∑ q ∈Q z ( β, q ) sin 2 π ( φ, q ) ( cos 2 π (∇ G, n q ) − ) q ( y )⎞⎠ Y ( φ ) and(3.7) ∂ y X x ( φ ) = − ⎛⎝ Q x ( y, φ ) +
12 2 π ∑ q ∈Q z ( β, q ) sin 2 π ( φ, q ) ( cos 2 π (∇ G x , n q ) − ) q ( y )⎞⎠ X x ( φ ) . We also define the random charges n Q x according to the formula(3.8) n Q x ∶= ∑ q ∈Q πz ( β, q ) ( cos 2 π ( φ, q ) sin 2 π (∇ G x , n q )) n q so that d n Q x = Q x . We note that by the exponential decay ∣ z ( β, q )∣ ≤ Ce − c √ β ∥ q ∥ , the decay of the gradient of the Green’s functionstated in (2.2) and the inequality ∣ sin a ∣ ≤ ∣ a ∣ , the random charges Q x and n Q x satisfy the L ∞ ( µ β ) -estimate:for each y ∈ Z d ,(3.9) ∥ Q x ( y, ⋅)∥ L ∞ ( µ β ) ≤ C ∣ y − x ∣ d − and ∥ n Q x ( y, ⋅)∥ L ∞ ( µ β ) ≤ C ∣ y − x ∣ d − . By a similar argument, but this time using the inequality ∣ cos a − ∣ ≤ ∣ a ∣ , one obtains the inequality, foreach y ∈ Z d ,(3.10) RRRRRRRRRRR ∑ q ∈Q z ( β, q ) sin 2 π ( φ, q ) ( cos 2 π (∇ G x , n q ) − ) q ( y )RRRRRRRRRRR ≤ C ∣ y − x ∣ d − and(3.11) RRRRRRRRRRR ∑ q ∈Q z ( β, q ) sin 2 π ( φ, q ) ( cos 2 π (∇ G x , n q ) − ) n q ( y )RRRRRRRRRRR ≤ C ∣ y − x ∣ d − . The reason we record the inequalities (3.10) and (3.11) is that, since 2 d − > d −
1, the function x ↦ ∣ x ∣ d − decays faster than x ↦ ∣ x ∣ d − . From this observation, we expect that the terms Q ( y ) Y and Q x ( y ) X x are theleading order terms in the identities (3.6) and (3.7) and that the terms involving the cosine of the gradient ofthe Green’s functions (left sides of (3.10) and (3.11)) are lower order terms which can be removed from theanalysis. We prove this result in the following lemma. Lemma 2.1 (Removing the contributions of the cosines) . One has the identity (3.12) cov [ X x , Y ] = ∑ y ∈ Z d ⟨ X x Q x ( y )V( y, ⋅)⟩ µ β + O ( C ∣ x ∣ d − − ε ) , where V is the solution of the Helffer-Sj¨ostrand equation, for each pair ( y, φ ) ∈ Z d × Ω , LV( y, φ ) = Q ( y, φ ) Y ( φ ) . A consequence of the identity (3.12) is the equivalence ∃ c ∈ R , cov [ X x , Y ] = c ∣ x ∣ d − + O ( C ∣ x ∣ d − + γ ′ ) ⇐⇒ ∃ c ∈ R , ∑ y ∈ Z d ⟨ X x Q x ( y )V( y, ⋅)⟩ µ β = c ∣ x ∣ d − + O ( C ∣ x ∣ d − + γ ′ ) . . DECOUPLING THE EXPONENTIALS 49 Remark . We recall that ε is the regularity exponents for the gradient and the mixed derivativesof the Green’s function G and G der stated in Proposition 4.7 of Chapter 3 and Proposition 4.2 of Chapter 5respectively.The proof of this result is again technical and does not represent the core of the argument; it is thusdeferred to Chapter 8. The argument relies on two ingredients:(i) We use the decay estimates for the Green’s function associated to the Helffer-Sj¨ostrand operator L and its mixed derivative stated in Proposition 4.7 of Chapter 3;(ii) We use the decay estimates (3.9) and (3.11) and take advantage of the fact that the function x ↦ ∣ x ∣ d − decays faster than the map x ↦ ∣ x ∣ d − .We complete this section by recording that we may also prove(3.13) ∃ c , c ∈ R , ⟨ Y Y x ⟩ µ β = c + c ∣ x ∣ d − + O ( C ∣ x ∣ d − + γ ′ ) by showing(3.14) cov [ Y x , Y ] = c ∣ x ∣ d − + O ( C ∣ x ∣ d − + γ ′ ) . Indeed, we have the following analogue of (3.12)cov [ Y x , Y ] = ∑ y ∈ Z d ⟨ Y x Q x ( y )V( y, ⋅)⟩ µ β + O ( C ∣ x ∣ d − − ε ) . The proof of this identity is almost the same as (3.12) with only notational changes, and is therefore omitted.
4. Decoupling the exponentials
The next (and final) technical step consists in removing the exponential terms X x and Y from thecomputation. To this end, we prove the decorrelation estimate stated in the following lemma. Lemma 3.1 (Decoupling the exponential terms) . One has the following estimate (4.1) cov [ X x , Y ] = ⟨ Y ⟩ µ β ⟨ X ⟩ µ β ∑ y ∈ Z d ⟨ Q x ( y, ⋅)U( y, ⋅)⟩ µ β + O ( C ∣ x ∣ d − + ε ) , and (4.2) cov [ Y x , Y ] = ⟨ Y ⟩ µ β ∑ y ∈ Z d ⟨ Q x ( y, ⋅)U( y, ⋅)⟩ µ β + O ( C ∣ x ∣ d − + ε ) . where the function U is the solution of the Helffer-Sj¨ostrand equation LU = Q in Z d × Ω . The identity (4.1) implies the equivalence ∃ c ∈ R , ∑ y ∈ Z d ⟨ X x Q x ( y, ⋅)V( y, ⋅)⟩ µ β = c ∣ x ∣ d − + O ( C ∣ x ∣ d − + γ ′ )⇐⇒ ∃ c ∈ R , ∑ y ∈ Z d ⟨ Q x ( y )U( y, ⋅)⟩ µ β = c ∣ x ∣ d − + O ( C ∣ x ∣ d − + γ ′ ) . Remark . The function U can be decomposed according to the following procedure: if for each charge q ∈ Q , we denote by U q the solution of the Helffer-Sj¨ostrand equation(4.3) LU q = cos 2 π ( φ, q ) q in Z d × Ω , then we have the identity(4.4) U = π ∑ q ∈Q z ( β, q ) sin 2 π (∇ G, n q )U q . Remark . By writing q = d n q , we can rewrite the equation (4.3) in the following form LU q = d ( cos 2 π (⋅ , q ) n q ) in Z d × Ω . As a consequence the function U q can be expressed in terms of the Green’s function G cos 2 π (⋅ ,q ) according tothe formula, for each pair ( y, φ ) ∈ Z d × Ω,(4.5) U q ( y, φ ) = ∑ z ∈ supp n q d ∗ z G cos 2 π (⋅ ,q ) ( y, φ ; z ) n q ( z ) . Using the decay estimate on the gradient and mixed derivative of the Green’s function given in Proposition 4.7of Chapter 3, we obtain that the map U q satisfies the upper bounds, for each y ∈ Z d ,(4.6) ∥U q ( y, ⋅)∥ L ∞ ( µ β ) ≤ C q ∣ y − z ∣ d − − ε and ∥∇U q ( y, ⋅)∥ L ∞ ( µ β ) ≤ C q ∣ y − z ∣ d − ε , where z is a point which belongs to the support of the charge n q (chosen arbitrarily). Remark . A consequence of the estimate (4.6) is that by using the exponential decay of the coefficient z ( β, q ) (see (1.10) of Chapter 3) and the inequality, for each charge q ∈ Q , ∣ sin 2 π (∇ G, n q )∣ ≤ π ∣(∇ G, n q )∣ ≤ π ∥∇ G ∥ L ( supp n q ) ∥ n q ∥ ≤ C q ∣ z ∣ d − , where z is a point in the support of n q (chosen arbitrarily), we deduce the inequality, for each point y ∈ Z d , ∥U( y, ⋅)∥ L ∞ ( µ β ) ≤ π ∑ z ∈ Z d ∑ q ∈Q z ∣ z ( β, q ) sin 2 π (∇ G, n q )∣ ∥U q ( y, ⋅)∥ L ∞ ( µ β ) ≤ ∑ z ∈ Z d ∑ q ∈Q z e − c √ β ∥ q ∥ C q ∣ z ∣ d − × ∣ y − z ∣ d − − ε ≤ C ∑ z ∈ Z d ∣ z ∣ d − × ∣ y − z ∣ d − − ε ≤ C ∣ y ∣ d − − ε . where we used the exponential decay of the term e − c √ β ∥ q ∥ to absorb the algebraic growth of the term C q ≤ C ∥ q ∥ k in the third inequality. The same argument also yields the estimate ∥∇U( y, ⋅)∥ L ∞ ( µ β ) ≤ C ∣ y ∣ d − − ε . We now give an heuristic argument explaining why we expect the decoupling estimate (4.1) to hold.
Heuristic of the proof of Lemma 3.1.
The strategy of the proof is to first decouple the exponentialterm X x and then decouple the exponential term Y ; to decouple the term X x , we prove the expansion(4.7) ∑ y ∈ Z d ⟨ X x Q x ( y, ⋅)V( y, ⋅)⟩ µ β = ⟨ X ⟩ µ β ∑ y ∈ Z d ⟨ Q x ( y, ⋅)V( y, ⋅)⟩ µ β + O ( C ∣ x ∣ d − − ε ) . A heuristic reason justifying why one can expect the expansion (4.7) to hold is the following. By the definitionof the random variable X x given in (1.4) and the decay of the gradient of the Green’s function ∇ G x statedin (2.2), we expect the random variable X x to essentially depend on the value of the gradient of the fieldaround the point x . The statement is voluntarily vague; one could give a mathematical meaning to it arguingthat if one considers a large constant C depending only on the dimension d , then the conditional expectationof the random variable X x with respect to the sigma-algebra generated by the fields (∇ φ ( y )) y ∈ B ( x,C ) is a goodapproximation of the random variable X x in the space L ( µ β ) .Additionally, using similar arguments to the one presented in Remarks 4.2 and 4.3, but using the L ( µ β ) -estimate ∥ Y ∥ L ( µ β ) ≤ C instead of the (stronger) pointwise upper bound ∣ cos 2 π ( φ, q )∣ ≤
1, one obtains the L ( µ β ) -estimate, for each y ∈ Z d ,(4.8) ∥∇V( y, ⋅)∥ L ( µ β ) ≤ C ∣ y ∣ d − − ε . . FIRST ORDER EXPANSION OF THE TWO-POINT FUNCTION 51 While we can prove the estimate (4.8) using Proposition 4.7 of Chapter 3, we expect that its real decay is oforder ∣ y ∣ − d , and make this assumption for the rest of the argument. We use an integration by parts to write,for each field φ ∈ Ω, ∑ y ∈ Z d Q x ( y, φ )V( y, φ ) = ∑ y ∈ Z d n Q x ( y, φ ) d ∗ V( y, φ ) . Since we expect the random charge n Q x ( y ) to decay like ∣ y − x ∣ − d (see the estimate (3.9)) and the randomvariable d ∗ V( y, ⋅) to decay ∣ y ∣ − d (since the codifferential d ∗ is a linear functional of the gradient ∇ ), we have(4.9) ∑ y ∈ Z d n Q x ( y, φ ) d ∗ V( y, φ ) ≃ ∑ y ∈ Z d ∣ y − x ∣ d − × ∣ y ∣ d − ≃ ∣ x ∣ d − . The point of the identity (4.9) is that while we expect the sum ∑ y ∈ Z d n Q x ( y, φ ) d ∗ V( y, φ ) to be of order ∣ x ∣ − d ,its restriction to the ball B ( x, C ) is of lower-order since we have ∑ y ∈ B ( x,C ) n Q x ( y, φ ) d ∗ V( y, φ ) ≃ ∑ y ∈ B ( x,C ) ∣ y − x ∣ d − × ∣ y ∣ d − ≃ ∣ x ∣ d − . A consequence of this result is that we expect the main contribution of the sum ∑ y ∈ Z d n Q x ( y, φ ) d ∗ V( y, φ ) tocome mostly from the points y outside the ball B ( x, C ) .To summarize the heuristic explanation, one should expect that: ● The random variable X x depends mostly on the gradient of the field inside a ball B ( x, C ) for somelarge but fixed constant C depending only on the dimension; ● The random variable ∑ y ∈ Z d n Q x ( y, φ ) d ∗ V( y, φ ) depends mostly on the value of the gradient of thefield outside the ball B ( x, C ) .Since the gradient of the field decorrelates, we expect the random variable ∑ y ∈ Z d n Q x ( y, φ ) d ∗ V( y, φ ) and X x to decorrelate; this is what is proved by (4.7).Once we have proved that the identity (4.7), we can prove the expansion (3.12) by applying the sameargument and the symmetry of the Helffer-Sj¨ostrand operator L to decorrelate the random variable Y .The previous paragraph describes a heuristic argument explaining why the expansion (4.1) is plausible; therigorous proof of the result is technical and deferred to Chapter 8. It relies on the Helffer-Sj¨ostrand formula: ifwe fix a point y ∈ Z d , then we can write ⟨ X x Q x ( y )V( y, ⋅)⟩ µ β − ⟨ X ⟩ µ β ⟨ Q x ( y )V( y, ⋅)⟩ µ β = cov [ X x , V( y, ⋅)] = ∑ z ∈ Z d ⟨X x ( z, ⋅) ∂ z V( y, ⋅)⟩ µ β , where the function X x is the solution of the Helffer-Sj¨ostrand equation, for each pair ( z, φ ) ∈ Z d × Ω, LX x ( z, φ ) = ∂ z X x ( φ ) . The map X x ( z, φ ) can then be explicitly written in terms of the Green’s function associated to the Helffer-Sj¨ostrand operator L described in Proposition 4.7 of Chapter 3. The derivative ∂ z V( y, ⋅) is studied usingproperties of the Green’s function associated to the differentiated Helffer-Sj¨ostrand operator L der statedin Proposition 4.2 of Chapter 5. The strategy relies on the fact that, since we have assumed the inversetemperature β to be large, a C , − ε -regularity theory holds for the operators L and L der , for some smallregularity exponent ε ∶= ε ( d, β ) >
0. If the exponent ε is small enough (or equivalently, if β is chosen largeenough), this regularity theory is precise enough to describe the behaviour of the Green’s functions G , G der accurately and we are able to prove that the absolute value of the covariance cov [ X x , V( y, ⋅)] is bounded fromabove by the value C ∣ x ∣ − d + ε , which gives the expansion (4.7).The expansion (4.1) can then be deduced from (4.7) by applying the same argument and using thesymmetry of the Helffer-Sj¨ostrand operator L . The expansion (4.2) can be proved by the same argument. (cid:3)
5. First order expansion of the two-point function
Once the Lemmas 1.1, 2.1 and 3.1 are established, we have showed that, to prove Theorem 3, it is enoughto obtain the expansion(5.1) ∃ c ∈ R , ∑ y ∈ Z d ⟨ Q x ( y )U( y, ⋅)⟩ µ β = c ∣ x ∣ d − + O ( C ∣ x ∣ d − + γ ′ ) . This section is devoted to the proof of (5.1). We first give a sketch of the proof in Section 5.1 and providethe details of the argument in Section 5.2.
In this section, we present a heuristic argument for the proof of the expan-sion (5.1). A large part of of the proof is concerned with the treatment of the technicalities inherent to thedual Villain model (sum over all the charges q ∈ Q , presence of a sine etc.). In order to highlight the mainideas of the argument, we make the following simplifications: ● We assume that for β large enough, one may essentially reduce the charges to the collection of dipoles ( d { y,y + e i } ) y ∈ Z d , ≤ i ≤ d . The exponential decay on the coefficient z ( β, q ) constraints the L -norm ofthe charge q to be small. One can thus assume that only the charges q ∈ Q which minimize thevalue ∥ q ∥ are involved in the sum; this leads us to considering the dipoles ( d { x,x + e i } ) x ∈ Z d , ≤ i ≤ d .An important, but mostly technical, part of the argument presented in Section 5.2 is devoted toproving that this dipole approximation yields the correct picture. Under this assumption, one hasthe simplifications Q x = d ∑ i = ∑ y ∈ Z d π sin ( π ∇ i G ( y )) d { y,y + e i } in U = d ∑ i = ∑ y ∈ Z d π sin ( π ∇ i G x ( y )) U y,i , where the function U y,i is the solution of the Helffer-Sj¨ostrand equation LU y,i = d ( cos 2 π ( d ∗ φ ( y ) ⋅ e i ) { y,y + e i } ) in Z d × Ω . ● Since the gradients of the Green’s functions ∇ i G ( y ) are usually small, we consider the first-orderexpansion of the sine and replace the value sin ( π ∇ i G x ( y )) by 2 π ∇ i G x ( y ) . With this assumption,we have Q x = ( π ) d ∑ i = ∑ y ∈ Z d ∇ i G ( y ) d { y,y + e i } and U = ( π ) d ∑ i = ∑ y ∈ Z d ∇ i G x ( y )U y,i . Using these simplifications, we compute(5.2) ∑ y ∈ Z d ⟨ Q x ( y )U( y, ⋅)⟩ µ β = ( π ) d ∑ i,j = ∑ y,y ∈ Z d ∇ i G ( y )∇ j G x ( y ) ⟨ cos ( d ∗ φ ( y ) ⋅ e i ) d ∗ U y,j ( y , φ ) ⋅ e i ⟩ µ β . Using the translation invariance of the measure µ β , one has the the identity, for each pair of points y, y ∈ Z d ,(5.3) ⟨ cos ( d ∗ φ ( y ) ⋅ e i ) d ∗ U y,j ( y , φ ) ⋅ e i ⟩ µ β = ⟨ cos ( d ∗ φ ( y − y ) ⋅ e i ) d ∗ U ,j ( y − y, φ ) ⋅ e i ⟩ µ β . Putting the identity (5.3) into the equality (5.2) and performing the change of variable κ ∶= y − y , we obtain(5.4) ∑ y ∈ Z d ⟨ Q x ( y )U( y, ⋅)⟩ µ β = ( π ) d ∑ i,j = ∑ y,κ ∈ Z d ∇ i G ( y )∇ j G x ( κ − y ) ⟨ cos ( d ∗ φ ( κ ) ⋅ e i ) d ∗ U ,j ( κ, φ ) ⋅ e i ⟩ µ β . The strategy is then to simplify the right side of (5.4) by arguing that the term d ∗ U ,j behaves like the mixedderivative of a deterministic Green’s function. Proving a quantitative version of this result is the subject ofTheorem 2 which is proved in Chapters 7 and 8; in this setting, it can be stated as follows: there exists anexponent γ ∶= γ ( d ) > i, j ∈ { , . . . , d } , there exists a deterministic constants c i,j ∶= c i,j ( d, β ) such that for each radius R ≥ ∑ κ ∈ B R ∖ B R RRRRRRRRRRR⟨ cos ( d ∗ φ ( κ ) ⋅ e i ) d ∗ U ,j ( κ, φ ) ⋅ e i ⟩ µ β − d ∑ i ,j = c i,i c j,j ∇ i ∇ j G ( κ )RRRRRRRRRRR ≤ CR γ . Once equipped with this estimate, we let E i,j ∶ Z d ↦ R be the error term defined according to the formula, foreach κ ∈ Z d , E i,j ( κ ) ∶= ⟨ cos ( d ∗ φ ( κ ) ⋅ e i ) d ∗ U ,j ( κ, φ ) ⋅ e i ⟩ µ β − d ∑ i ,j = c i,i c j,j ∇ i ∇ j G ( κ ) . According to the regularity estimate on the gradient of the Green’s matrix associated to the Helffer-Sj¨ostrandoperator L stated in Proposition 4.7 of Chapter 3 (via the formula (4.5)) and the homogenization estimate (5.5),this term satisfies the L and pointwise estimates(5.6) ∀ R ≥ , R − d ∑ κ ∈ B R ∖ B R ∣E i,j ( κ )∣ ≤ CR d + γ and ∀ κ ∈ Z d , ∣E i,j ( κ )∣ ≤ C ∣ κ ∣ d − ε . . FIRST ORDER EXPANSION OF THE TWO-POINT FUNCTION 53 We can use the definition of the term E i,j to rewrite the identity (5.4). We obtain ∑ y ∈ Z d ⟨ Q x ( y )U( y, ⋅)⟩ µ β = ( π ) d ∑ i,i ,j,j = c i,i c j,j ∑ y,κ ∈ Z d ∇ i G ( y )∇ j G x ( κ − y )∇ i ∇ j G ( κ ) (5.7) + d ∑ i,j = ∇ i G ( y )∇ j G x ( κ − y )E i,j ( κ ) . The right side of the identity (5.7) can then be refined. First using the estimates (5.6) on the error term E i,j and Proposition 5.1 proved in Section 5 of Chapter 8, we can show the following expansion: there exists anexponent γ ′ ∶= γ ′ ( d ) > d ∑ i,j = ∑ y,κ ∈ Z d ∇ i G ( y )∇ j G x ( κ − y )E i,j ( κ ) = d ∑ i,j = K i,j ∑ y,κ ∈ Z d ∇ i G ( y )∇ j G x ( κ − y ) + O ( C ∣ x ∣ d − + γ ′ ) , where the constant K i,j are obtained from the error term E i,j according to the formula K i,j ∶= ( π ) ∑ κ ∈ Z d E i,j ( κ ) , which, by the estimate (5.6), is well-defined. A combination of the identity (5.7) with the expansion (5.8) thenshows ∑ y ∈ Z d ⟨ Q x ( y )U( y, ⋅)⟩ µ β = π d ∑ i,i ,j,j = c i,i c j,j ∑ y,κ ∈ Z d ∇ i G ( y )∇ j G x ( κ − y )∇ i ∇ j G ( κ ) (5.9) + d ∑ i,j = K i,j ∑ y,κ ∈ Z d ∇ i G ( y )∇ j G x ( κ − y ) + O ( C ∣ x ∣ d − + γ ′ ) . This expansion does not give the result (5.1) directly and we need to exploit the symmetries of the dual Villainmodel to conclude. The argument relies on the following observation: since the Villain and dual Villain modelare invariant under the action of the group H of the lattice preserving transformations introduced in Chapter 2,the same property holds for the two-point function and thus for the map x ↦ ∑ y ∈ Z d ⟨ Q x ( y )U( y, ⋅)⟩ µ β .One can then use this invariance property together with the expansion (5.9) to prove that this expansionmust take the simpler form(5.10) ∑ y ∈ Z d ⟨ Q x ( y )U( y, ⋅)⟩ µ β = c ∣ x ∣ d − + O ( C ∣ x ∣ d − + γ ′ ) . This is achieved by using the property of the discrete Green’s function and relies on tools from Fourier analysis.The proof can be found in Section 4 of Chapter 8. The expansion (5.10) is exactly (5.1); the proof is thuscomplete.
We first write Q x = d n Q x , perform an integration by parts and use the identities (3.8)and (4.4) to expand the sum ∑ y ∈ Z d ⟨ Q x ( y )U( y, ⋅)⟩ µ β . We obtain ∑ y ∈ Z d ⟨ Q x ( y )U( y, ⋅)⟩ µ β (5.11) = ∑ y ∈ Z d ⟨ n Q x ( y ) d ∗ U( y, ⋅)⟩ µ β = π ∑ y ∈ Z d ∑ q ,q ∈Q z ( β, q ) z ( β, q ) sin 2 π (∇ G, n q ) sin 2 π (∇ G x , n q ) ⟨ cos 2 π ( φ, q ) d ∗ U q ( y, φ )⟩ µ β n q ( y ) . To simplify the sum over all the charges q , q , we introduce an equivalence class on the set of charges Q : wesay that two charges q and q ′ are equivalent, and denote it by q ∼ q ′ , if and only if one is the translation of theother, i.e., q ∼ q ′ ⇐⇒ ∃ z ∈ Z d , q ( z + ⋅) = q ′ . We denote this quotient space by Q/ Z d and for each charge q ∈ Q , we denote by [ q ] its equivalence class. Foreach equivalence class [ q ] ∈ Q/ Z d , we select a charge q ∈ Q such that 0 belongs to the support of n q (if there is more than one candidate, we break ties by using an arbitrary criterion). We note that, for each charge q ∈ Q ,by the definition of the charge n q and the coefficient z ( β, q ) , we have the identities, for each point z ∈ Z d ,(5.12) z ( β, q ) = z ( β, q (⋅ − z )) , n q (⋅− z ) = n q (⋅ − z ) and ( n q (⋅− z ) ) = ( n q ) . We also note that, by using the translation invariance of the measure µ β and the definition of the function U q given in (4.3), we have the equality, for each pair of points ( y, z ) ∈ Z d , ⟨ cos 2 π ( φ, q ) d ∗ U q (⋅− z ) ( y, φ )⟩ µ β = ⟨ cos 2 π ( φ, q (⋅ + z )) d ∗ U q ( y − z, φ )⟩ µ β . Additionally, we can decompose the sum over the charges q ∈ Q along the equivalence classes, i.e., we can write,for any summable function F ∶ Q → R ,(5.13) ∑ q ∈Q F ( q ) = ∑ [ q ]∈Q/ Z d ∑ z ∈ Z d F ( q (⋅ − z )) , where the charge q in the right side is the element of the equivalence class [ q ] ∈ Q/ Z d chosen such that 0belongs to the support of the charge n q .Combining the identities (5.12) and (5.13), we can rewrite the equality (5.11),(5.14) ∑ y ∈ Z d ⟨ Q x ( y )U( y, ⋅)⟩ µ β = π ∑ [ q ] , [ q ]∈Q/ Z d z ( β, q ) z ( β, q )×⎡⎢⎢⎢⎢⎣ ∑ z ,z ,y ∈ Z d sin 2 π (∇ G, n q (⋅ − z )) sin 2 π (∇ G x , n q (⋅ − z )) ⟨ cos 2 π ( φ, q (⋅ − z + z )) d ∗ U q ( y − z , φ )⟩ µ β n q ( y − z )⎤⎥⎥⎥⎥⎦ . We first rearrange the identity (5.14). We use the identities (∇ G x , n q (⋅ − z )) = (∇ G x (⋅ + z ) , n q ) , (∇ G, n q (⋅ − z )) = (∇ G (⋅ + z ) , n q ) and perform the change of variable y ∶= y − z . We obtain(5.15) ∑ y ∈ Z d ⟨ Q x ( y )U( y, ⋅)⟩ µ β = π ∑ [ q ] , [ q ]∈Q/ Z d z ( β, q ) z ( β, q )×⎡⎢⎢⎢⎢⎣ ∑ z ,z ,y ∈ Z d sin 2 π (∇ G (⋅ + z ) , n q ) sin 2 π (∇ G x (⋅ + z ) , n q ) ⟨ cos 2 π ( φ, q (⋅ − z + z )) d ∗ U q ( y + z − z , φ )⟩ µ β n q ( y )⎤⎥⎥⎥⎥⎦ . ·„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„¶ (5.15) −( q ,q ) The rest of the proof is decomposed into two steps:(1) In the first step, we use Theorem 2 and the regularity estimates established in Proposition 4.7of Chapter 3 to prove the following result: there exists an exponent γ ′ ∶= γ ′ ( d ) > q , q ∈ Q and each pair of integers ( i, j ) ∈ { , . . . , d } , there exist constants K q ,q ∶= K q ,q ( q , q , d, β ) , C q ,q ∶= C q ,q ( q , q , d, β ) , c q ij ∶= c q ij ( i, j, q , d, β ) such that the term(5.15) − ( q , q ) satisfies the expansion(5.16) (5.15) − ( q , q ) = d ∑ i,j,k,l = c q ij c q kl ∑ z ,z ∈ Z d ∇ i G ( z )∇ j ∇ k G ( z − z )∇ l G ( x − z )+ K q ,q ∑ z ∈ Z d ∇ G ( z ) ⋅ ( n q ) × ∇ G x ( x − z ) ⋅ ( n q ) + O ( C q ,q ∣ x ∣ d − + γ ′ ) . We recall that the vectors ( n q ) and ( n q ) belongs to R d and are defined by the formulas ( n q ) ∶= ∑ y ∈ Z d n q ( y ) and ( n q ) ∶= ∑ y ∈ Z d n q ( y ) . These two quantities belong to the space R d (or more precisely Z d ). We also record that all theconstants K q ,q , c q ,q ij and C q ,q grow at most algebraically fast in the values ∥ q ∥ and ∥ q ∥ ,i.e., there exist an exponent k ∶= k ( d ) < ∞ and a constant C ∶= C ( d, β ) < ∞ such that one has theestimates(5.17) ∣ c q ij ∣ ≤ C ∥ q ∥ k , ∣ K q ,q ∣ ≤ C ∥ q ∥ k ∥ q ∥ k and ∣ C q ,q ∣ ≤ C ∥ q ∥ k ∥ q ∥ k . (2) In the second step, we use the symmetry and rotation invariance of the dual Villain model to provethat the expansion (5.16) implies the expansion (5.1). . FIRST ORDER EXPANSION OF THE TWO-POINT FUNCTION 55 We first give the proof of the second item of (5.16), as the argument is simpler and less technical. We firstsum the expansion (5.16) over all the equivalence classes [ q ] , [ q ] ∈ Q/ Z d and use the exponential decay ofthe coefficients z ( β, q ) and z ( β, q ) to absorb the algebraic growth of the constants c q ij , c q ij and C q ,q . Weobtain ∑ y ∈ Z d ⟨ Q x ( y )U( y, ⋅)⟩ µ β = π ∑ [ q ] , [ q ]∈Q/ Z d z ( β, q ) z ( β, q ) × (5.15) − ( q , q ) (5.18) = π ∑ [ q ] , [ q ]∈Q/ Z d d ∑ i,j,k,l = z ( β, q ) z ( β, q ) c q ij c q kl ∑ z ,z ∈ Z d ∇ i G ( z )∇ j ∇ k G ( z − z )∇ l G ( x − z )+ π ∑ [ q ] , [ q ]∈Q/ Z d z ( β, q ) z ( β, q ) K q ,q ∑ z ∈ Z d ∇ G ( z ) ⋅ ( n q ) × ∇ G ( x − z ) ⋅ ( n q )+ π ∑ [ q ] , [ q ]∈Q/ Z d z ( β, q ) z ( β, q ) O ( C q ,q ∣ x ∣ d − + γ ′ )= d ∑ i,j,k,l = c ij c kl ∑ z ,z ∈ Z d ∇ i G ( z )∇ j ∇ k G ( z − z )∇ l G ( x − z )+ d ∑ i,j = K i,j ∑ z ∈ Z d ∇ i G ( z )∇ j G ( x − z ) + O ( C ∣ x ∣ d − + γ ′ ) , where we have set c ij ∶= π ∑ [ q ]∈Q/ Z d z ( β, q ) c qij and K i,j ∶= π ∑ [ q ] , [ q ]∈Q/ Z d z ( β, q ) z ( β, q ) K q ,q [( n q ) ⋅ e i ] × [( n q ) ⋅ e j ] , which are well-defined by the estimate ∣ z ( β, q )∣ ≤ e − c √ β ∥ q ∥ and (5.17).We then simplify the expansion (5.18) by noting that, since the measure µ β is invariant under thesymmetries and rotations of the lattice Z d , the function x ↦ ∑ y ∈ Z d ⟨ Q x ( y )U( y, ⋅)⟩ µ β is also invariant over thesymmetries and rotations of the lattice. It is proved in Proposition 5.1 in Chapter 8 that this invarianceproperty combined with the expansion (5.18) implies that there exists a constant c ∶= c ( d, β ) such that ∑ y ∈ Z d ⟨ Q x ( y )U( y, ⋅)⟩ µ β = c ∣ x ∣ d − + O ( C ∣ x ∣ d − + γ ′ ) . This is precisely the expansion (5.1). We have thus proved that the expansion (5.16) implies the expansion (5.1).The rest of the demonstration is devoted to the proof of (5.16). We first simplify the term (5.15) − ( q , q ) by removing the sine. To this end, we use the following ingredients: ● We use the inequality, ∣ sin a − a ∣ ≤ a , valid for any real number a ∈ R and the inequality, for eachcharge q ∈ Q and each point z ∈ Z d , ∣(∇ G, n q (⋅ − z ))∣ ≤ ∥∇ G ( z + ⋅)∥ L ( supp n q ) ∥ n q ∥ ≤ C q ∣ z ∣ d − . We deduce that, for each pair of charges q , q ∈ Q and each pair of points z , z ∈ Z d ,(5.19) ∣ sin 2 π (∇ G x (⋅ + z ) , n q ) − π (∇ G (⋅ + z ) , n q )∣ ≤ C q ∣ z ∣ d − and(5.20) ∣ sin 2 π (∇ G x (⋅ + z ) , n q ) − π (∇ G x (⋅ + z ) , n q )∣ ≤ C q ∣ z − x ∣ d − ; ● We further simplify the terms 2 π (∇ G, n q (⋅ − z )) and 2 π (∇ G x , n q (⋅ − z )) . We use that the doublegradient of the Green’s function decays like ∣ z ∣ − d and the assumption that 0 belongs to the support of the charges n q and n q to write ∣ π (∇ G x (⋅ + z ) , n q ) − π ( n q ) ⋅ ∇ G x ( z )∣ = ∣ π (∇ G x ( z + ⋅) − ∇ G x ( z ) , n q )∣ (5.21) ≤ C ∥ n q ∥ ∥∇ G x ( z + ⋅) − ∇ G x ( z )∥ L ( supp n q ) ≤ C ∥ n q ∥ ∣ supp n q ∣ sup z ∈ supp n q ∣∇∇ G ( z + z − x )∣≤ C q ∣ z − x ∣ d . A similar argument shows the estimate(5.22) ∣ π (∇ G, n q (⋅ − z )) − π ( n q ) ⋅ ∇ G ( z )∣ ≤ C q ∣ z ∣ d . We then combine the inequalities (5.19) and (5.21) on the one hand and (5.20) and (5.22) on theother hand and use the inequality 3 d − > d . We obtain the two estimates(5.23) ∣ sin 2 π (∇ G x , n q (⋅ − z )) − π ( n q ) ⋅ ∇ G x ( z )∣ ≤ C q ∣ x − z ∣ d and(5.24) ∣ sin 2 π (∇ G x , n q (⋅ − z )) − π ( n q ) ⋅ ∇ G ( z )∣ ≤ C q ∣ z ∣ d ; ● We recall the estimate (4.6) which reads for each y ∈ Z d , ∥∇U q ( y, ⋅)∥ L ∞ ( µ β ) ≤ C q ∣ x − y ∣ d − ε . From thisinequality, we deduce that for each pair of charges q , q ∈ Q such that 0 belongs to the supports of n q and n q and for each point y ∈ supp n q , ∣⟨ cos 2 π ( φ, q (⋅ − z + z )) d ∗ U q ( y + z − z , φ )⟩ µ β ∣ ≤ C q ,q ∣ z − z ∣ d − ε . ● We have the inequalities, for each point x ∈ Z d ,(5.25) ∑ z ,z ∈ Z d ∣ x − z ∣ d × ∣ z − z ∣ d − ε × ∣ z ∣ d − ≤ C ln ∣ x ∣∣ x ∣ d − − ε and ∑ z ,z ∈ Z d ∣ x − z ∣ d − × ∣ z − z ∣ d − ε × ∣ z ∣ d ≤ C ∣ x ∣ d − − ε . The proof of these estimates are postponed to Proposition 0.2 of Appendix C.A combination of the the four items described above implies the following expansion(5.26) (5.15) − ( q , q )= π ∑ z ,z ,y ∈ Z d ∇ G ( z ) ⋅ ( n q )∇ G x ( z ) ⋅ ( n q ) ⟨ cos 2 π ( φ, q (⋅ − z + z )) d ∗ U q ( y + z − z , φ )⟩ µ β n q ( y )·„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„¶ (5.26) −( q ,q ) + O ( C q ,q ∣ x ∣ d − − ε ) . A consequence of the identity (5.26) is that to prove the expansion (5.16), it is enough to prove the followingresult(5.27) (5.26) − ( q , q ) = d ∑ i,j,k,l = c q ij c q kl ∑ z ,z ∈ Z d ∇ i G ( z )∇ j ∇ k G ( z − z )∇ l G ( x − z )+ π K q ,q ∑ z ,y ∈ Z d ∇ G ( z ) ⋅ ( n q )∇ G x ( z + κ ) ⋅ ( n q ) + O ( C q ,q ∣ x ∣ d − + γ ) . The rest of the argument is devoted to the proof of (5.27) and relies on the homogenization of the mixedderivative associated to the Helffer-Sj¨ostrand equation (Theorem 2). . FIRST ORDER EXPANSION OF THE TWO-POINT FUNCTION 57
We first consider the term (5.26) − ( q , q ) and perform the change of variable κ ∶= z − z . We obtain(5.28) (5.26) − ( q , q )= π ∑ z ,κ,y ∈ Z d ∇ G ( z ) ⋅ ( n q )∇ G x ( z + κ ) ⋅ ( n q ) ⟨ cos 2 π ( φ, q (⋅ − κ )) d ∗ U q ( y + κ, φ )⟩ µ β n q ( y ) . We then apply the homogenization theorem for the solution of the Helffer-Sj¨ostrand equation stated inTheorem 2.We post-process the result of Theorem 2 so that it can be used to estimate the term (5.26) − ( q , q ) ; theobjective is to prove the estimate (5.36) below. We first use that the codifferential d ∗ is a linear functional ofthe gradient to deduce from Theorem 2 that, for each radius R ≥ XXXXXXXXXXXXX d ∗ U q − ∑ ≤ i ≤ d ∑ ≤ j ≤( d ) ( d ∗ l e ij + d ∗ χ ij ) ∇ i G q ,j XXXXXXXXXXXXX L ( B R ∖ B R ,µ β ) ≤ C q R d + γ . To ease the notations, we let A R ∶= B R ∖ B R . Using the arguments and notations introduced in Remark (4.9)of Chapter 3, we obtain the identity ∑ ≤ i ≤ d ∑ ≤ j ≤( d ) ( d ∗ l e ij + d ∗ χ ij ) ∇ i G q ,j = ∑ ≤ i ≤ d ( e i + d ∗ χ i ) ( d ∗ G q ⋅ e i ) . The estimate (5.29) then implies, by using the stationarity of the gradient of the infinite-volume corrector andthe Cauchy-Schwarz inequality(5.30) 1 R d ∑ κ ∈ A R ∣⟨ cos 2 π ( φ, q (⋅ − κ )) d ∗ U q ( κ, φ )⟩ µ β − ∑ ≤ i ≤ d ⟨ cos 2 π ( φ, q ) ( e i + d ∗ χ i ( , φ ))⟩ µ β ( d ∗ G q ( κ ) ⋅ e i )∣≤ C q R d + γ . We then generalize the inequality (5.30) and prove the following result: for each point y ∈ Z d , one has theestimate(5.31)1 R d ∑ κ ∈ A R ∣⟨ cos 2 π ( φ, q (⋅ − κ )) d ∗ U q ( y + κ, φ )⟩ µ β − ∑ ≤ i ≤ d ⟨ cos 2 π ( φ, q ) ( e i + d ∗ χ i ( y, φ ))⟩ µ β ( d ∗ G q ( κ ) ⋅ e i )∣≤ C q ( + ∣ y ∣ d + γ ) R d + γ . To prove this result, we distinguish two cases, whether the norm of y is larger or smaller than R . Case 1. The norm of y is smaller than R . In that case, we note that the set y + A R is included in theannuli B R ∖ B R . We can use the identity (5.29) with the two annulus A R and A R to deduce that(5.32)1 R d ∑ κ ∈ A R ∣⟨ cos 2 π ( φ, q (⋅ − κ )) d ∗ U q ( y + κ, φ )⟩ µ β − ∑ ≤ i ≤ d ⟨ cos 2 π ( φ, q ) ( e i + d ∗ χ i ( y, φ ))⟩ µ β ( d ∗ G q ( y + κ ) ⋅ e i )∣≤ C q R d + γ . We then use the definition of the function G q stated in (4.19) combined with the fact that the triple gradientof the Green’s function G decays like z ↦ ∣ z ∣ d + . We deduce that, for each point κ ∈ A R ,(5.33) ∣( d ∗ G q ( y + κ ) ⋅ e i ) − ∇ i ( d ∗ G q ( κ ) ⋅ e i )∣ ≤ ∣ y ∣ × supp z ∈ B ( κ,y ) ∣∇ G q ( κ )∣ ≤ C q ∣ y ∣ R d + . A combination of the inequalities (5.32) and (5.33) implies the estimate (5.31) in Case 1.
Case 2. The norm of y is larger than R . In that case, we use the following (rough) estimates: for eachpair of points y, κ ∈ Z d ,(5.34) ∣⟨ cos 2 π ( φ, q (⋅ − κ )) d ∗ U q ( y + κ, φ )⟩ µ β ∣ ≤ C q and(5.35) ∑ ≤ i ≤ d ∣⟨ cos 2 π ( φ, q ) ( e i + d ∗ χ i ( y, φ ))⟩ µ β ( d ∗ G q ( y + κ ) ⋅ e i )∣ ≤ C q . Using the estimates (5.34) and (5.35), the fact that the volume of the annulus A R is of order R d and theassumption ∣ y ∣ ≥ R , we deduce that ∑ κ ∈ A R ∣⟨ cos 2 π ( φ, q (⋅ − κ )) d ∗ U q ( y + κ, φ )⟩ µ β − ∑ ≤ i ≤ d ⟨ cos 2 π ( φ, q ) ( e i + d ∗ χ i ( y, φ ))⟩ µ β ( d ∗ G q ( y + κ ) ⋅ e i )∣≤ C q R d ≤ d + γ C q ∣ y ∣ d + γ R d + γ . The proof of the estimate (5.31) is complete.We then consider the estimate (5.31) for a point y in the support of the charge n q , consider the scalarproduct with the vector n q ( y ) and the sum over all the points y in the support of n q . We additionally usethe inequalities ∣ n q ∣ ≤ C q , ∣ supp n q ∣ ≤ C q and the fact that, since 0 belongs to the support of n q , one hasthe estimate, for each point y in the support of n q , ∣ y ∣ ≤ diam n q ≤ C q . We obtain ∑ κ ∈ A R ∣⟨ cos 2 π ( φ, q (⋅ − κ )) n q ( y ) d ∗ U q ( y + κ, φ )⟩ µ β − ∑ ≤ i ≤ d ⟨ cos 2 π ( φ, q ) n q ( y ) ( e i + d ∗ χ i ( y, φ ))⟩ µ β ( d ∗ G q ( κ ) ⋅ e i )∣≤ C q ,q R γ . We sum over all the points y in the support of the charge n q and obtain(5.36) ∑ κ ∈ A R ∣⟨ cos 2 π ( φ, q (⋅ − κ )) ( n q , d ∗ U q (⋅ + κ, φ ))⟩ µ β − ∑ ≤ i ≤ d ⟨ cos 2 π ( φ, q ) ( n q , e i + d ∗ χ i ( y, φ ))⟩ µ β ( d ∗ G q ( κ ) ⋅ e i )∣≤ C q ,q R γ . We then focus on the term (5.26) − ( q , q ) (and more specifically on the right side of (5.28)) and use theinequality (5.36) to simplify it. To ease the notation, we introduce the following definitions: ● We let E q ,q be the map from Z d to R defined according to the formula, for each point κ ∈ Z d , E q ,q ( κ ) ∶= ⟨ cos 2 π ( φ, q (⋅ − κ )) ( n q , d ∗ U q (⋅ + κ, φ ))⟩ µ β − ∑ ≤ i ≤ d ⟨ cos 2 π ( φ, q ) ( n q , e i + d ∗ χ i )⟩ µ β ( d ∗ G q ( κ ) ⋅ e i ) , It is an error term which is small; in view of the estimate (5.36), Remark 4.2 and the definition ofthe map G q ,j stated in (4.19) of Theorem 2, it satisfies the inequalities(5.37) ∀ R ≥ , ∑ κ ∈ A R ∣E q ,q ( κ )∣ ≤ C q ,q R γ and ∀ κ ∈ Z d , ∣E q ,q ( κ )∣ ≤ C q ,q ∣ κ ∣ d − ε ; ● We recall the definition of the coefficient λ β stated in Remark 4.2 of Chapter 3. For each pair ofintegers i, j ∈ { , . . . , d } , we define the coefficient c qij according to the formula c qij ∶= π ( + λ β ) − [( n q ) ⋅ e i ] × ⟨ cos 2 π ( φ, q ) ( n q , d ∗ l e ij + d ∗ χ ij )⟩ µ β . Using these notations together with Remark 4.9 of Chapter 3 and an explicit computation (which we omithere), we obtain the formula ∑ ≤ i ≤ d ⟨ cos 2 π ( φ, q ) ( n q , e i + d ∗ χ i ( y, φ ))⟩ µ β ( d ∗ G q ( κ ) ⋅ e i )= ( + λ β ) − ∑ ≤ i,j ≤ d ⟨ cos 2 π ( φ, q ) ( n q , e i + d ∗ χ i )⟩ µ β ⟨ cos 2 π ( φ, q ) ( n q , e i + d ∗ χ i )⟩ µ β ∇ i ∇ j G. . FIRST ORDER EXPANSION OF THE TWO-POINT FUNCTION 59 The term (5.26) − ( q , q ) then becomes(5.26) − ( q , q ) = d ∑ i,j,k,l = c q ij c q kl ∑ z ,z ∈ Z d ∇ i G ( z )∇ j ∇ k G ( z − z )∇ l G ( x − z )+ π ∑ z ,κ ∈ Z d ∇ G ( z ) ⋅ ( n q )∇ G ( z + κ − x ) ⋅ ( n q )E q ,q ( κ ) . To prove the estimate (5.27), it is thus sufficient to prove that there exists a constant K q ,q such that one hasthe expansion4 π ∑ z ,κ ∈ Z d ∇ G ( z ) ⋅ ( n q )∇ G x ( z + κ ) ⋅ ( n q )E q ,q ( κ )= K q ,q ∑ z ∈ Z d ∇ G ( z ) ⋅ ( n q )∇ G x ( z ) ⋅ ( n q ) + O ( C q ,q R d + γ ′ ) . The proof of this result relies on the estimates (5.37); it is the subject of Proposition 5.1 and is deferred toChapter 8. We note that the argument gives an explicit value for the constant K q ,q which is given by K q ,q = π ∑ κ ∈ Z d E q ,q ( κ ) . By the estimates (5.37), the constant K q ,q is well-defined and grows at most algebraically fast in theparameters ∥ q ∥ and ∥ q ∥ as required.HAPTER 5 Regularity theory for low temperature dual Villain model
This chapter is devoted to the study of the solutions of the Helffer-Sj¨ostrand operator(0.1)
L ∶= ∆ φ − β ∆ + β ∑ n ≥ β n (− ∆ ) n + + ∑ q ∈Q ∇ ∗ q ⋅ a q ∇ q , where we recall the notation, for each integer-valued closed and compactly supported charge q ∈ Q ,(0.2) ∇ ∗ q ⋅ a q ∇ q u = z ( β, q ) cos 2 π ( φ, q ) ( u, q ) q. This operator acts on the space Z d × Ω. We decompose it into two terms: the operator ∆ φ which acts on thefield φ and the spatial term L spat defined by the formula L spat ∶= − β ∆ + β ∑ n ≥ β n (− ∆ ) n + + ∑ q ∈Q ∇ ∗ q ⋅ a q ∇ q . The operator L spat is uniformly elliptic. The purpose of this chapter is to apply the techniques from the theoryof elliptic regularity to understand the behavior of the solutions of the equations L spat u = L u =
0. Westudy three types of objects: ● In Sections 1 and 2, we study the solutions of the equation L u =
0. We establish a Caccioppoliinequality (Proposition 1.1) and C , − ε -regularity estimates (Proposition 2.4); ● In Section 3, we study the Green’s matrix G f and the heat kernel P f . We prove Gaussian bounds onthe heat kernel, decay estimates on the Green’s matrix and C , − ε -regularity estimates; ● Section 4 is devoted to the study of the Green’s matrix G der , f and the heat kernel P der , f associatedto the differentiated Helffer-Sj¨ostrand L der (see (4.5)); we prove Gaussian bounds on the heat kerneland decay estimates on the Green’s matrix and C , − ε -regularity estimates.Let us give a few comments and heuristic of the proofs presented in this chapter. The demonstrations relyon two main ingredients: ● If we write(0.3)
L = ∆ φ − β ∆ ·„„„„„„„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„„„„„„„¶ L + β ∑ n ≥ β n (− ∆ ) n + + ∑ q ∈Q ∇ ∗ q ⋅ a q ∇ q ·„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„¶ L pert , then the operator L pert is a perturbation of L if the inverse temperature β is large enough. Theoperator L has properties similar to the Laplacian and a complete regularity theory is available.The strategy to obtain regularity estimates relies on Schauder theory (see [ , Chapter 3]): sincethe operator L pert is a perturbation of the operator L , one can prove that each solution of theequation L u = u of the equation L u =
0. One canthen borrow the regularity of the function u and transfer it to the function u . This process causesa deterioration of the regularity for the function u : one obtains a C , − ε -regularity theory for thesolutions of the system L u =
0, for some strictly positive exponent ε . The size of the exponent ε depends on the inverse temperature β and tends to 0 as β tends to infinity. ● The second ingredient is the Feynman-Kac formula which is used to study heat kernel and Green’smatrix and is described in the following paragraph. Given a real number p ∈ [ , ∞) and a function f ∈ L p ( µ β ) , we let P f be the solution of the parabolic equation(0.4) ⎧⎪⎪⎨⎪⎪⎩ ∂ t P f + LP f = ( , ∞) × Ω × Z d , P f ( , ⋅ , ⋅) = f (⋅) δ in Ω × Z d .
612 5. REGULARITY THEORY FOR LOW TEMPERATURE DUAL VILLAIN MODEL
Given a field φ ∈ Ω which satisfies the growth condition ∑ x ∈ Z d ∣ φ ( x )∣ e − r ∣ x ∣ < ∞ for some r >
0, we let ( φ t ) t ≥ be the diffusion process evolving according to the Langevin dynamics(0.5) ⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩ dφ t ( x ) = β ∆ φ t ( x ) dt − β ∑ n ≥ β n (− ∆ ) n + φ t ( x ) dt + ∑ q ∈Q (∇ ∗ q ⋅ a q ( φ t )∇ q φ t ) ( x ) dt + dB t ( x ) ,φ ( x ) = φ ( x ) , where ( B t ( x )) x ∈ R d is a collection of normalized R ( d ) -valued independent Brownian motions. Wedenote by P φ the law of the dynamics ( φ t ) t ≥ starting from φ and by E φ the expectation with respectto the measure P φ . The solvability of the SDE (0.5) is guaranteed by standard arguments (see [ ,Section 2.1.3] or [ , Section 2.2]). The solution P f of the parabolic system (0.4) is related to thediffusion process ( φ t ) t ≥ through the Feynman-Kac representation formula which reads as follows,for each ( x, φ ) ∈ Z d × Ω, one has the identity(0.6) P f ( t, x, φ ; y ) = E φ [ f ( φ t ) P φ ⋅ ( t, x, y )] , where P φ ⋅ is the solution of the parabolic system ⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩ ∂ t P φ ⋅ ( t, ⋅ , y ) − β ∆ P φ ⋅ ( t, ⋅) + β ∑ n ≥ β n (− ∆ ) n + P φ ⋅ ( t, ⋅) + ∑ q ∈Q (∇ ∗ q ⋅ a q ( φ t )∇ q P φ ⋅ ( t, ⋅)) = Z d ,P φ ⋅ ( , ⋅ , y ) = δ y in Z d . The rigorous justification of the Feynman-Kac formula requires to use tools from spectral theory.The argument in the case of the dual Villain model is identical to the one presented for the uniformlyelliptic ∇ φ -model which can be found in the articles of Naddaf and Spencer [ ] and Giacomin, Ollaand Spohn [ ] and not presented here.While most of the ideas and techniques come from the theory of elliptic and parabolic regularity, one needsto treat the infinite range of the elliptic operator L spat ; for each integer n ∈ N , the iteration of the Laplacian∆ n has range 2 n and the sum over all the charges q ∈ Q involves charges with arbitrarily large support.Nevertheless one has exponential decay for the long range terms, due to the exponent β n and to the estimate(1.10) of Chapter 3 on the coefficient z ( β, q ) ; this allows to prove that the contribution of these terms isnegligible and to obtain the desired results.We complete this section by mentioning that we need to keep track of the dependence of the constants inthe inverse temperature β ; our objective is to prove that the regularity exponent ε tends to 0 as the inversetemperature β tends to infinity. We thus only allow the constants to depend on the dimension.
1. Caccioppoli inequality for the solutions of the Helffer-Sj¨ostrand equation
In this section, we prove a Caccioppoli inequality for the operator L , the proof follows the standardtechnique but some technical difficulties have to be taken into account due the infinite range of the operator L .In particular, the result obtained is slightly different from the one of the standard Caccioppoli inequality: thereis a long range term in the right sides of (1.2) and (1.3) which takes into consideration the infinite range ofthe Helffer-Sj¨ostrand operator. Since the coefficients of the operator L decay exponentially fast, the long rangeterm in the right sides of (1.2) and (1.3) exhibits the same decay. Proposition . Fix a radius R ≥ and let u ∶ Z d → R ( d ) be a solution of theHelffer-Sj¨ostrand equation (1.1) L u = in B R × Ω . Then there exist a constant C ∶= C ( d ) < ∞ and an exponent c ∶= c ( d ) > such that the following estimates hold (1.2) β ∑ y ∈ Z d ∥ ∂ y u ∥ L ( B R ,µ β ) + ∥∇ u ∥ L ( B R ,µ β ) ≤ CR ∥ u ∥ L ( B R ,µ β ) + ∑ x ∈ Z d ∖ B R e − c ( ln β )∣ x ∣ ∥ u ( x, ⋅)∥ L ( µ β ) , and (1.3) ∥∇ u ∥ L ( B R ,µ β ) ≤ CR ∥ u − ( u ) B R ∥ L ( B R ,µ β ) + ∑ x ∈ Z d e − c ( ln β )( R ∨∣ x ∣) ∥ u ( x, ⋅)∥ L ( µ β ) . . CACCIOPPOLI INEQUALITY FOR THE SOLUTIONS OF THE HELFFER-SJ ¨OSTRAND EQUATION 63 Remark . The two long range terms in the right sides of (1.2) and (1.3) are error terms which are smalland are caused by the infinite range of the operator L spat . They decay exponentially fast and are typically oforder e − R . Proof.
The strategy of the proof follows the standard outline of the proof of the Caccioppoli inequality.We let η ∶ Z d → R be a cutoff function satisfying the following properties(1.4) 0 ≤ η ≤ , η = B R , η = Z d ∖ B R and ∣∇ η ∣ ≤ CR .
We note that the properties on the function η imply, for each charge q ∈ Q ,(1.5) sup supp n q η ≤ inf supp n q η + C diam n q R ≤ inf supp n q η + C q R .
By testing the function ηu in the equation (1.1), we obtain(1.6) β ∑ x,y ∈ Z d η ( x ) ⟨( ∂ y u ( x, φ )) ⟩ µ β + ∑ x ∈ Z d ⟨∇ u ( x, φ ) ⋅ ∇ ( η u ) ( x, φ )⟩ µ β + β ∑ q ∈Q ⟨∇ q u ⋅ a q ∇ q ( η u )⟩ µ β ·„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„¶ (1.6) −( i ) + ∑ n ≥ ∑ x ∈ Z d β n ⟨∇ n + u ( x, φ ) ⋅ ∇ n + ( η u ) ( x, φ )⟩ µ β ·„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„¶ (1.6) −( ii ) = . The terms in the first line can be estimated with the standard arguments of the Caccioppoli inequality. Theterms in the second line require specific considerations.To estimate the term (1.6)-(i), we first fix a charge q ∈ Q and write ∇ q ( η u ) = ( η u, q ) = ( d ∗ ( η u ) , n q ) . We then expand the codifferential of the product d ∗ ( η u ) thanks to the formula (2.10) of Chapter 2. Weobtain d ∗ ( η u ) ( x ) = L k, d ∗ (∇ ( η u ) ( x, φ )) = L k, d ∗ ( η ( x )∇ u ( x, φ )) + L k, d ∗ (∇ η ( x ) ⊗ u ( x, φ ))= η ( x ) L k, d ∗ (∇ u ( x, φ )) + L k, d ∗ ( η ( x )∇ η ( x ) ⊗ u ( x, φ ))= η ( x ) d ∗ u ( x, φ ) + η ( x ) L k, d ∗ (∇ η ( x ) ⊗ u ( x, φ )) . We use the estimate (1.4) on the gradient of η and use that the linear map L k, d ∗ is bounded to deduce thefollowing inequality ∣∇ q ( η u ) − ( η d ∗ u, n q )∣ ≤ C ∥ n q ∥ L ∞ R ∑ z ∈ supp q ∣ η ( z ) u ( z, φ )∣ ≤ C q R ∑ z ∈ supp q ∣ η ( z ) u ( z, φ )∣ . We use the estimate (1.4) on the gradient of η a second time, the fact that the codifferential is a boundedoperator and the estimate (1.5), to obtain, for any point x in the support of the charge n q , ∣( η d ∗ u, n q ) − η ( x ) ( d ∗ u, n q )∣ = ∣(( η − η ( x )) d ∗ u, n q )∣ (1.7) ≤ ∣(( η − η ( x ))( η + η ( x )) d ∗ u, n q )∣≤ C q R ∑ z ∈ supp n q ∣( η ( z ) + η ( x )) u ( z, φ )∣≤ C q R ∑ z ∈ supp n q ∣ η ( z ) u ( z, φ )∣ + C q R ∑ z ∈ supp n q ∣ u ( z, φ )∣ . A combination of the two previous displays yields the inequality, for any point x ∈ supp n q ,(1.8) ∣∇ q ( η u ) − η ( x ) ∇ q u ∣ ≤ C q R ∑ z ∈ supp n q ∣ η ( z ) u ( z, φ )∣ + C q R ∑ z ∈ supp n q ∣ u ( z, φ )∣ . For each charge q ∈ Q , we select a point x q which belongs to its support arbitrarily. Applying the estimate (1.8)with the point x = x q , the Cauchy-Schwarz inequality and the estimate(1.9) ∣ a q ∣ ≤ ∣ z ( β, q )∣∥ q ∥ ≤ C q e − c √ β ∥ q ∥ , we obtain ⟨∇ q u ⋅ a q ∇ q ( η u )⟩ µ β ≥ η ( x q ) ⟨∇ q u ⋅ a q ∇ q u ⟩ µ β − C q e − c √ β ∥ q ∥ R ∥∇ q u ∥ L ( µ β ) ∑ x ∈ supp n q η ( x ) ∥ u ( x, ⋅)∥ L ( µ β ) − C q e − c √ β ∥ q ∥ R ∥∇ q u ∥ L ( µ β ) ∑ x ∈ supp n q ∥ u ( x, ⋅)∥ L ( µ β ) . We sum over all the charges q ∈ Q such that the support of η intersects the support of n q . We obtain theestimate ∑ q ∈Q ⟨∇ q u ⋅ a q ∇ q ( η u )⟩ µ β (1.10) ≥ ∑ q ∈Q η ( x q ) ⟨∇ q u ⋅ a q ∇ q u ⟩ µ β ·„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„¶ (1.10) −( i ) − ∑ q ∈Q C q e − c √ β ∥ q ∥ R ∥∇ q u ∥ L ( µ β ) ∑ z ∈ supp n q η ( z ) ∥ u ( z, ⋅)∥ L ( µ β ) ·„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„¶ (1.10) −( ii ) − ∑ q ∈Q C q e − c √ β ∥ q ∥ { supp n q ∩ supp η ≠∅} R ∥∇ q u ∥ L ( µ β ) ∑ x ∈ supp n q ∥ u ( x, ⋅)∥ L ( µ β ) ·„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„¶ (1.10) −( iii ) . We then estimate the three terms in the right side separately. The term (1.10)-(i) can be estimated thanks tothe estimate (1.9) on the value of the coefficient a q . We obtain ∑ q ∈Q η ( x q ) ∣⟨∇ q u ⋅ a q ∇ q u ⟩ µ β ∣ ≤ ∑ q ∈Q C q e − c √ β ∥ q ∥ η ( x q ) ∣⟨( d ∗ u, n q ) ⟩ µ β ∣≤ ∑ q ∈Q C q e − c √ β ∥ q ∥ η ( x q ) ∑ x ∈ supp n q ∥∇ u ( x, ⋅)∥ L ( µ β ) ≤ ∑ q ∈Q C q e − c √ β ∥ q ∥ × ⎛⎝ ∑ x ∈ supp n q η ( x ) ∥∇ u ( x, ⋅)∥ L ( µ β ) + ∑ x ∈ supp n q ( η ( x q ) − η ( x ) ) ∥∇ u ( x, ⋅)∥ L ( µ β ) ⎞⎠ . We use the estimates (1.4) and (1.5) on the function η and a computation similar to the one performed in (1.7)and the fact that R is always larger than 1. We obtain(1.11) ∑ q ∈Q η ( x q ) ∣⟨∇ q u ⋅ a q ∇ q u ⟩ µ β ∣≤ ∑ q ∈Q C q e − c √ β ∥ q ∥ ⎛⎝ ∑ x ∈ supp n q η ( x ) ∥∇ u ( x, ⋅)∥ L ( µ β ) + R ∑ x ∈ supp n q { supp n q ∩ supp η ≠∅} ∥∇ u ( x, ⋅)∥ L ( µ β ) ⎞⎠ . We then use that, since the support of the function η is included in the ball B R , one has the inequalities, foreach point x ∈ Z d ,(1.12) ⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩ ∑ q ∈Q x C q { supp n q ∩ supp η ≠∅} e − c √ β ∥ q ∥ ≤ Ce − c √ β dist ( x,B R ) , ∑ q ∈Q x C q e − c √ β ∥ q ∥ ≤ Ce − c √ β . These estimates are a consequence of the inequalities (2.7) in Chapter 2. Using (1.12) and the fact that thediscrete gradient is a bounded operator, we can simplify the estimate (1.11) and obtain(1.13) ∑ q ∈Q η ( x q ) ⟨∇ q u ⋅ a q ∇ q u ⟩ µ β ≥ − Ce − c √ β ∑ x ∈ Z d η ( x ) ∥∇ u ( x, ⋅)∥ L ( µ β ) − CR ∑ x ∈ Z d e − c √ β dist ( x,B R ) ∥ u ( x, ⋅)∥ L ( µ β ) . This completes the estimate of the term (1.10)-(i). . CACCIOPPOLI INEQUALITY FOR THE SOLUTIONS OF THE HELFFER-SJ ¨OSTRAND EQUATION 65
To estimate the term (1.10)-(ii), we fix a charge q ∈ Q . We note that only the charges q ′ such that thesupport of n q ′ intersects the support of η (or equivalently the ball B R ) are counted in the sum; we can thusassume without loss of generality that the support of n q intersects the support of η . We have the inequality,for each field φ ∈ Ω, ∣∇ q u (⋅ , φ )∣ = ( d ∗ u (⋅ , φ ) , n q ) ≤ ∥ n q ∥ ∞ ∑ z ∈ supp q ∣∇ u ( z, φ )∣ ≤ C q ∑ z ∈ supp q ∣∇ u ( z, φ )∣ . Using this estimate, we obtain ∥∇ q u ∥ L ( µ β ) ∑ z ∈ supp n q η ( z ) ∥ u ( z, ⋅)∥ L ( µ β ) ≤ C q ⎛⎝ ∑ z ∈ supp n q ∥∇ u ( z, ⋅)∥ L ( µ β ) ⎞⎠ ⎛⎝ ∑ z ∈ supp n q η ( z ) ∥ u ( z, ⋅)∥ L ( µ β ) ⎞⎠≤ C q ( sup supp n q η ) ⎛⎝ ∑ z ∈ supp n q ∥∇ u ( z, ⋅)∥ L ( µ β ) ⎞⎠ ⎛⎝ ∑ z ∈ supp n q ∥ u ( z, ⋅)∥ L ( µ β ) ⎞⎠ . We then use the property (1.5) to deduce that ∥∇ q u ∥ L ( µ β ) ∑ z ∈ supp n q η ( z ) ∥ u ( z, ⋅)∥ L ( µ β ) (1.14) ≤ C q ( inf supp n q η + C q R ) ⎛⎝ ∑ z ∈ supp n q ∥∇ u ( z, ⋅)∥ L ( µ β ) ⎞⎠ ⎛⎝ ∑ z ∈ supp n q ∥ u ( z, ⋅)∥ L ( µ β ) ⎞⎠≤ C q ⎛⎝ ∑ z ∈ supp n q η ( z ) ∥∇ u ( z, ⋅)∥ L ( µ β ) ⎞⎠ ⎛⎝ ∑ z ∈ supp n q ∥ u ( z, ⋅)∥ L ( µ β ) ⎞⎠+ C q R ⎛⎝ ∑ z ∈ supp n q ∥∇ u ( z, ⋅)∥ L ( µ β ) ⎞⎠ ⎛⎝ ∑ z ∈ supp n q ∥ u ( z, ⋅)∥ L ( µ β ) ⎞⎠ . We estimate the first term of the right side of the inequality (1.14) by the Cauchy-Schwarz and Young’sinequalities(1.15) ⎛⎝ ∑ z ∈ supp n q η ( z ) ∥∇ u ( z, ⋅)∥ L ( µ β ) ⎞⎠ ⎛⎝ ∑ z ∈ supp n q ∥ u ( z, ⋅)∥ L ( µ β ) ⎞⎠≤ C q R ∑ z ∈ supp n q η ( z ) ∥∇ u ( z, ⋅)∥ L ( µ β ) + C q R ∑ z ∈ supp n q ∥ u ( z, ⋅)∥ L ( µ β ) . To estimate the second term in the right side of (1.14), we use that the discrete gradient is a bounded operatorand the Cauchy-Schwarz inequality. We obtain(1.16) ⎛⎝ ∑ z ∈ supp n q ∥∇ u ( z, ⋅)∥ L ( µ β ) ⎞⎠ ⎛⎝ ∑ z ∈ supp n q ∥ u ( z, ⋅)∥ L ( µ β ) ⎞⎠ ≤ C q ∑ z ∈ supp n q ∥ u ( z, ⋅)∥ L ( µ β ) . Collecting the estimates (1.14), (1.15), (1.16), we obtain the upper bound for the term (1.10)-(ii)(1.17) ∥∇ q u ∥ L ( µ β ) ∑ z ∈ supp n q η ( z ) ∥ u ( z, ⋅)∥ L ( µ β ) ≤ C q R ∑ z ∈ supp n q η ( z ) ∥∇ u ( z, ⋅)∥ L ( µ β ) + C q R ∑ z ∈ supp n q ∥ u ( z, ⋅)∥ L ( µ β ) . Multiplying the inequality (1.17) by e − c √ β ∥ q ∥ , summing over all the charges q ∈ Q and using the estimates (1.12),we obtain(1.18) 1 R ∑ q ∈Q C q e − c √ β ∥ q ∥ ∥∇ q u ∥ L ( µ β ) ∑ z ∈ supp n q η ( z ) ∥ u ( z, ⋅)∥ L ( µ β ) ≤ Ce − c √ β ∑ x ∈ Z d η ( x ) ∥∇ u ( x, ⋅)∥ L ( µ β ) + CR ∑ x ∈ Z d e − c √ β dist ( x,B R ) ∥ u ( x, ⋅)∥ L ( µ β ) . The term (1.10)-(iii) is estimated following a similar strategy and we omit the details. We obtain(1.19) ∑ q ∈Q C q e − c √ β ∥ q ∥ { supp n q ∩ supp η ≠∅} R ∥∇ q u ∥ L ( µ β ) ∑ x ∈ supp n q ∥ u ( x, ⋅)∥ L ( µ β ) ≤ Ce − c √ β ∑ x ∈ Z d η ( x ) ∥∇ u ( x, ⋅)∥ L ( µ β ) + CR ∑ x ∈ Z d e − c √ β dist ( x,B R ) ∥ u ( x, ⋅)∥ L ( µ β ) . Combining the estimates (1.10), (1.13), (1.18), (1.19) and choosing the inverse temperature β large enoughso that the exponential terms Ce − c √ β are smaller than β , we obtain the inequality(1.20) β ∑ q ∈Q ⟨∇ q u ⋅ a q ∇ q ( η u )⟩ µ β ≥ − ∑ x ∈ Z d η ( x ) ∥∇ u ( x, ⋅)∥ L ( µ β ) − CR ∑ x ∈ Z d e − c √ β dist ( x,B R ) ∥ u ( x, ⋅)∥ L ( µ β ) . This completes the estimate of the term (1.6)-(i).There only remains to estimate the term (1.6)-(ii) pertaining to the iterations of the Laplacian. The prooffollows similar ideas so we do not write down every details. We use the two following ingredients: the discretegradient is an operator which has a finite operator norm in L ( Z d ) and the discrete operator ∇ n has range n .By expanding the term ∇ n ( ηu ) , and using the estimate on the gradient of η stated in (1.4), we obtain thefollowing inequality, for each pair ( x, φ ) ∈ Z d × Ω,(1.21) ∣∇ n ( η u ) ( x, φ ) − η ( x )∇ n u ( x, φ )∣ ≤ C n R ∑ z ∈ B ( x,n ) η ( z )∣ u ( z, φ )∣ . Using this inequality, we obtain(1.22) ∑ x ∈ B R ⟨∇ n u ( x, ⋅) ⋅ ∇ n ( η u ) ( x, ⋅)⟩ µ β ≥ ∑ x ∈ B R η ( x ) ∥∇ n u ( x, ⋅)∥ L ( µ β ) − C n R ∑ x ∈ B R ∥∇ n u ( x, ⋅)∥ L ( µ β ) ∑ z ∈ B ( x,n ) η ( z ) ∥ u ( z, ⋅)∥ L ( µ β ) . We then use that the discrete gradient is a bounded operator, the properties on the function η stated in (1.4)and the fact that the volume of the ball B n is of order n d (and has smaller growth than the exponential term C n ). We obtain the estimate(1.23) ∑ x ∈ B R ∥∇ n u ( x, ⋅)∥ L ( µ β ) ∑ z ∈ B ( x,n ) η ( z ) ∥ u ( z, ⋅)∥ L ( µ β ) ≤ R C n ⎛⎝ ∑ x ∈ B R η ( x ) ∥∇ n u ( x, ⋅)∥ L ( µ β ) ⎞⎠ + ̃ C n R ∑ x ∈ B R + n ∥ u ( x, ⋅)∥ L ( µ β ) , where C is the constant in (1.22) and ̃ C > C . Combining the estimates (1.22) and (1.23), we obtain(1.24) ∑ x ∈ B R ⟨∇ n u ( x, ⋅) ⋅ ∇ n ( η u ) ( x, ⋅)⟩ µ β ≥ ∑ x ∈ B R η ( x ) ∥∇ n u ( x, ⋅)∥ L ( µ β ) − C n R ∑ x ∈ B R + n ∥ u ( x, ⋅)∥ L ( µ β ) . Multiplying both sides of the inequality (1.24) by β n + and summing over the integers n ∈ N , we obtain theinequality12 β ∑ n ≥ β n ∑ x ∈ B R ⟨∇ n + u ( x, ⋅) ⋅ ∇ n + ( η u ) ( x, ⋅)⟩ µ β ≥ β ( ∑ n ≥ β n ) ∑ x ∈ B R η ( x ) ∥∇ n + u ( x, ⋅)∥ L ( µ β ) − R ∑ n ≥ ( C √ β ) n ∑ x ∈ B R + n ∥ u ( x, ⋅)∥ L ( µ β ) . . REGULARITY THEORY FOR THE HELFFER-SJ ¨OSTRAND OPERATOR 67 We use that the first term in the right side of (1.24) is non-negative and assume that the inverse temperature β is chosen large enough to rewrite12 β ∑ n ≥ β n ∑ x ∈ B R ⟨∇ n + u ( x, ⋅) ⋅ ∇ n + ( η u ) ( x, ⋅)⟩ µ β ≥ − R ∑ n ∈ N ( C √ β ) n ∑ x ∈ B R + n ∥ u ( x, ⋅)∥ L ( µ β ) ≥ − CR √ β ∑ x ∈ Z d ( C √ β ) dist ( x,B R ) ∥ u ( x, ⋅)∥ L ( µ β ) . If β is chosen large enough (for instance larger than C ) then there exists a constant c > C √ β ≤ e − c ( ln β ) . We have obtained(1.25) ∑ n ≥ β n ∑ x ∈ B R ⟨∇ n + u ( x, ⋅) ⋅ ∇ n + ( η u ) ( x, ⋅)⟩ µ β ≥ − CR √ β ∑ x ∈ Z d e − c ( ln β ) dist ( x,B R ) ∥ u ( x, ⋅)∥ L ( µ β ) . This completes the estimate of the term (1.6)-(ii).We then choose the inverse temperature β large enough, combine the identity (1.6), the esti-mates (1.20), (1.25) and the standard arguments of the proof of the Caccioppoli inequality. We obtainthe inequality(1.26) β ∑ y ∈ Z d ∥ ∂ y u ∥ L ( B R ,µ β ) + ∥∇ u ∥ L ( B R ,µ β ) ≤ CR ∥ u ∥ L ( B R ,µ β ) + CR + d ∑ x ∈ Z d ∖ B R e − c ( ln β ) dist ( x,B R ) ∥ u ( x, ⋅)∥ L ( µ β ) . We finally post-process the estimate (1.26) and use the inequality CR + d ∑ x ∈ Z d ∖ B R e − c ( ln β ) dist ( x,B R ) ∥ u ( x, ⋅)∥ L ( µ β ) (1.27) ≤ CR ∥ u ∥ L ( B R ) + C ∑ x ∈ Z d ∖ B R e − c ( ln β ) dist ( x,B R ) ∥ u ( x, ⋅)∥ L ( µ β ) ≤ CR ∥ u ∥ L ( B R ) + C ∑ x ∈ Z d ∖ B R e − c ( ln β )∣ x ∣ ∥ u ( x, ⋅)∥ L ( µ β ) , where in the second line we used the inequality ∣ x ∣ ≤ C dist ( x, B R ) valid for any point x ∈ Z d ∖ B R . Acombination of the estimates (1.26) and (1.27) shows the inequality(1.28) β ∑ y ∈ Z d ∥ ∂ y u ∥ L ( B R ,µ β ) + ∥∇ u ∥ L ( B R ,µ β ) ≤ CR ∥ u ∥ L ( B R ,µ β ) + C ∑ x ∈ Z d ∖ B R e − c ( ln β )∣ x ∣ ∥ u ( x, ⋅)∥ L ( µ β ) . This is the estimate (1.2) up to one difference: the right side of (1.28) involves the ball B R while the rightside of (1.2) involves the ball B R . This is a minor issue which can be fixed easily and we skip the details.The proof of the estimate (1.3) follows similar lines, except that we need to use the function η ( u − ∑ x ∈ Z d η ( x ) ∑ x ∈ Z d η ( x ) u ( x, ⋅)) as a test function in the equation (1.1). The proof is similar andthe same technicalities (pertaining to the iterations of the Laplacian and to the sum over the charges q ∈ Q )need to be treated; since the proof does not contain any new element, we omit the details. (cid:3)
2. Regularity theory for the Helffer-Sj¨ostrand operator
The purpose of this section is to prove the C , − ε -regularity of the solutions of the Helffer-Sj¨ostrandequation (0.1). The result is stated in Proposition 2.4.The proof relies on Schauder theory; as is explained in (0.3), the strategy is to decompose the Helffer-Sj¨ostrand operator L into two terms, denoted L and L pert as recalled below L = ∆ φ − β ∆ ·„„„„„„„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„„„„„„„¶ L + β ∑ n ≥ β n (− ∆ ) n + + ∑ q ∈Q ∇ ∗ q ⋅ a q ∇ q ·„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„¶ L pert . The operator L is the leading order term. For this operator a C , -regularity theory is available, similar tothe one of the Laplacian. This result is stated in Proposition 2.1; the proof is essentially equivalent to thestandard proof of the regularity of the Laplacian.The second operator L pert is a perturbation term; it is small when the inverse temperature β is large. Thestrategy is to argue that any solution u of the Helffer-Sj¨ostrand equation is well-approximated on every scaleby a function u of the equation ∆ φ u + β ∆ u = u to obtain a C , − ε -regularity estimate on the solution u . This section can be decomposed into three propositions: ● Proposition 2.1 establishes a regularity theory for the solutions u of the equation L u = ● Proposition 2.2 states that if a function u is well-approximated, in the sense of the estimate (2.3)below, by a solution of the equation ∆ φ u − β ∆ u =
0, then a C , − ε -regularity estimate holds for thefunction u ; ● Proposition 2.4 establishes the regularity for the solutions of the Helffer-Sj¨ostrand equation. Weprove that any solution u of the equation L u = u of the equation L u = ∆ φ − β ∆ . In this section, we establish a regularity theoryfor the operator ∆ φ − β ∆. Proposition φ − β ∆) . Fix a radius R > and let u ∶ B R × Ω be a solution of the equation ∆ φ u − β ∆ u = in B R × Ω . Then, for any integer k ∈ N , there exists a constant C k < ∞ depending on the dimension d and the integer k such that the following estimate holds (2.1) sup x ∈ B R ∥∇ k u ( x, ⋅)∥ L ( µ β ) ≤ C k R k + d ∥ u − ( u ) B R ∥ L ( B R ,µ β ) . Proof.
The proof is standard and relies on two ingredients: the Caccioppoli inequality and the observationthat the spatial gradient ∇ commutes with the two Laplacians ∆ φ and ∆. First by the Caccioppoli inequality,which can be deduced from the standard proof as explained in Section 1, one has ∥∇ u ∥ L ( B R ,µ β ) ≤ CR ∥ u − ( u ) B R ∥ L ( B R ,µ β ) . We then note that, since u is a solution of the equation L u =
0, the gradient of u is also a solution of theequation L ∇ u =
0. Once can thus apply the Caccioppoli inequality to the gradient of u and deduce ∥∇ u ∥ L ( B R ,µ β ) ≤ CR ∥∇ u ∥ L ( B R ,µ β ) . An iteration of this argument shows that, for any integer k ≥
1, the L ( B R , µ β ) -norm of the iterated gradient ∇ k u is controlled by the L ( B R , µ β ) -norm of the function u with the appropriate scaling. By an applicationof the Sobolev embedding theorem (see [ , Chapter 4]), we obtain the regularity estimate (2.1). (cid:3) The next proposition states that if amap u is well-approximated on every scale by a solution u of the equation L u =
0, then the function u satisfiesa C , − ε regularity estimate for some exponent ε depending only on the dimension d and the precision of theapproximation. The proof follows a well-known strategy of Campanato (see e.g. [ ]). The proof writtenbelow is an adaptation of the one of Hofmann and Kim [ ]. Proposition . Fix a radius X ≥ , a regularity exponent ε > and a constant K > . There existsa constant δ ε > , depending on the parameters d and ε such that for any radius R ≥ X , and any function u ∈ L ( B R , µ β ) satisfying the property that, for any radius r ∈ [ X, R ] , there exists a function u ∈ L ( B r , µ β ) ,solution of the equation (2.2) ∆ φ u − β ∆ u = in B r × Ω , satisfying (2.3) ∥∇( u − u )∥ L ( B r ,µ β ) ≤ δ ε ∥∇ u ∥ L ( B r ,µ β ) + CK, . REGULARITY THEORY FOR THE HELFFER-SJ ¨OSTRAND OPERATOR 69 then there exists a constant C ∶= C ( ε, d ) < ∞ such that for every r ∈ [ X, R ] , ∥∇ u ∥ L ( B r ,µ β ) ≤ C ( Rr ) ε ∥∇ u ∥ L ( B R ,µ β ) + K. Before starting the proof, we record the following lemma, which is a consequence of Giaquinta [ , Lemma2.1]. Lemma . Fix two non-negative real numbers
X, R such that R ≥ X ≥ and two non-negative constants C , K . For any regularity exponent ε > , there exist two constants δ ε ∶= δ ε ( C, ε, d ) and C ∶= C ( C, ε, d ) suchthat the following statement holds. If φ ∶ R + → R is a non-negative and non-decreasing function which satisfiesthe estimate, for each pair of real numbers ρ, r ∈ [ X, R ] satisfying ρ ≤ r , (2.4) φ ( ρ ) ≤ C ⎛⎝( ρr ) d + δ ε ⎞⎠ φ ( r ) + K, then one has the estimate, for any ρ, r ∈ [ X, R ] satisfying ρ ≤ r , (2.5) φ ( ρ ) ≤ C ⎛⎝( ρr ) d − ε φ ( r ) + Kρ d ⎞⎠ . Proof.
This lemma can be extracted from [ , Lemma 2.1 p86] by setting α = d , β = d − ε and by usingthat the radii R, r are larger than 1 . (cid:3) Proof of Proposition 2.2.
We fix a regularity exponent ε >
0, let δ ε > ρ, r ∈ [ X, R ] . We let u be the solution of the equation (2.2) in theset B r × Ω such that the estimate (2.3) holds. We note that the estimate (2.3) implies the inequality ∥∇ u ∥ L ( B r ,µ β ) ≤ C ∥∇ u ∥ L ( B r ) + K . By the regularity theory for the map u established in Proposition 2.1, wehave(2.6) ∥∇ u ∥ L ( B ρ ,µ β ) ≤ C ( ρr ) d ∥∇ u ∥ L ( B r ,µ β ) . By combining the estimates (2.3) and (2.6) and the estimate on the L -norm of the gradient of u mentionedabove, we compute ∥∇ u ∥ L ( B ρ ,µ β ) ≤ ∥∇ ( u − u )∥ L ( B ρ ,µ β ) + ∥∇ u ∥ L ( B ρ ,µ β ) ≤ ∥∇ ( u − u )∥ L ( B r ,µ β ) + ( ρr ) d ∥∇ u ∥ L ( B r ,µ β ) ≤ δ ε ∥∇ u ∥ L ( B r ,µ β ) + K + ( ρr ) d ( C ∥∇ u ∥ L ( B r ) + K )≤ C ⎛⎝( ρr ) d + δ ε ⎞⎠ ∥∇ u ∥ L ( B r ,µ β ) + K. We apply Lemma 2.3 with the function φ ( ρ ) = ∥∇ u ∥ L ( B ρ ) . The inequality (2.5) with the choice r = R gives,for any radius ρ ∈ [ X, R ] , ∥∇ u ∥ L ( B ρ ,µ β ) ≤ C ⎛⎝( ρR ) d − ε ∥∇ u ∥ L ( B R ,µ β ) + Kρ d ⎞⎠ . Dividing both side of the estimate by ρ d completes the proof. (cid:3) We now use Propositions 2.1 and 2.2 to obtain C , − ε -regularity for the solutions of the Helffer-Sj¨ostrandequation. Proposition C , − ε -regularity theory) . For any regularity exponent ε > , there exists an inversetemperature β ∶= β ( d, ε ) < ∞ such that the following statement holds. There exist two constants C ∶= C ( d, ε ) < ∞ and c ∶= c ( d ) > such that for any radius R ≥ , any inverse temperature β ≥ β and any function u ∶ Z d × Ω → R solution of the equation L u = in B R × Ω , one has the estimate (2.7) ∥∇ u ( , ⋅)∥ L ( µ β ) ≤ CR − ε ∥ u − ( u ) B R ∥ L ( B R ,µ β ) + ∑ x ∈ Z d e − c ( ln β )( R ∨∣ x ∣) ∥ u ( x, ⋅)∥ L ( µ β ) . Proof.
The strategy of the proof is to apply Proposition 2.2 to the function u and then to apply theCaccioppoli inequality. We fix a regularity exponent ε >
0, a radius R ≥ ● In Step 1, we prove that the map u satisfies the following property: there exist an inverse temperature β ( ε, d ) < ∞ and a constant C ∶= C ( d ) < ∞ such that for every β > β and every radius r ≥ ( ln R ) ,the following estimate holds(2.8) ∥∇ u ∥ L ( B r ,µ β ) ≤ C ( Rr ) ε ∥∇ u ∥ L ( B R ,µ β ) + ∑ x ∈ Z d ∖ B R e − c ( ln β )∣ x ∣ ∥∇ u ( x, ⋅)∥ L ( µ β ) . ● In Step 2, we deduce from (2.8) and the Caccioppoli inequality stated in Proposition 1.1, the pointwiseestimate (2.7).
Step 1.
To prove the estimate (2.8), the strategy is to apply Proposition 2.2. To this end, we set X ∶= ( ln R ) , and fix a radius r ∈ [ X, R ] . We then define the function u to be the solution of the boundaryvalue problem(2.9) ⎧⎪⎪⎪⎨⎪⎪⎪⎩ ∆ φ u − β ∆ u = B r × Ω ,u = u on ∂B r × Ω . We first prove that the map u is a good approximation of the map u . Specifically, we prove that there existtwo constants C ∶= C ( d ) < ∞ and c ∶= c ( d ) > ∥∇( u − u )∥ L ( B r ,µ β ) ≤ Cβ ∥∇ u ∥ L ( B r ,µ β ) + Ce − c ln β ( ln R ) ∥∇ u ∥ L ( B R ,µ β ) + C ∑ x ∈ Z d ∖ B ( ,R ) e − c ln β ∣ x ∣ ∥∇ u ( x, ⋅)∥ L ( µ β ) . To prove the estimate (2.10), we note that the map u − u is a solution of the following system of equations(2.11) ⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩ ∆ φ ( u − u ) − β ∆ ( u − u ) = − β ∑ n ≥ β n (− ∆ ) n + u − ∑ q ∈Q ∇ ∗ q ⋅ a q ∇ q u in B r × Ω ,u − u = ∂B r × Ω . We extend the function ( u − u ) by 0 outside the ball B R so that it is defined on the entire space Z d and use itas a test function in the system (2.11). We obtain(2.12) ∑ y ∈ Z d ∥ ∂ y ( u − u )∥ L ( B r ,µ β ) + β ∥∇ ( u − u )∥ L ( B r ,µ β ) = − β ∑ n ≥ β n ∑ x ∈ Z d ⟨∇ n + u ( x, ⋅) ⋅ ∇ n + ( u − u )( x, ⋅)⟩ µ β ·„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„¶ (2.12) −( i ) − ∑ q ∈Q ⟨∇ q u ⋅ a q ∇ q ( u − u )⟩ µ β ·„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„¶ (2.12) −( ii ) . We first focus on the term (2.12)-(i) and note that, for each integer n ∈ N , since the iterated gradient ∇ n hasrange n and since the map u − u is equal to 0 outside the ball B r , the function ∇ n ( u − u ) is supported in theball B r + n . By the Cauchy-Schwarz inequality, we obtain ∑ x ∈ Z d ⟨∇ n + u ( x, ⋅) , ∇ n + ( u − u )( x, ⋅)⟩ µ β ≤ ∥∇ n + u ∥ L ( B r + n ,µ β ) ∥∇ n + ( u − u )∥ L ( B r ,µ β ) . Since the discrete gradient ∇ has a finite operator norm on L ( Z d ) , one has the estimate ∑ x ∈ Z d ⟨∇ n + u ( x, ⋅) , ∇ n + ( u − u )( x, ⋅)⟩ µ β ≤ ∣∣∣∇ n ∣∣∣ L → L ∥∇ u ∥ L ( B r + n + ,µ β ) ∥∇( u − u )∥ L ( B r ,µ β ) (2.13) ≤ C n (∥∇ u ∥ L ( B r + n + ,µ β ) + ∥∇( u − u )∥ L ( B r ,µ β ) ) , . REGULARITY THEORY FOR THE HELFFER-SJ ¨OSTRAND OPERATOR 71 where we used the inequality of operator norms ∣∣∣∇ n ∣∣∣ L → L ≤ ∣∣∣∇∣∣∣ nL → L ≤ C n and Young’s inequality in thesecond line.Multiplying the inequality (2.13) by β − n and summing over all the integers n ∈ N , we obtain ∑ n ≥ β n ∑ x ∈ Z d ⟨∇ n + u ( x, ⋅) , ∇ n + ( u − u )( x, ⋅)⟩ µ β ≤ ∑ n ≥ C n β n ∥∇ u ∥ L ( B r + n ,µ β ) + ∑ n ≥ C n β n ∥∇( u − u )∥ L ( B r ,µ β ) . by choosing the inverse temperature β large enough (at larger than the square of the constant C ), we obtain(2.14) ∑ n ≥ β n ∑ x ∈ Z d ⟨∇ n + u ( x, ⋅) , ∇ n + ( u − u )( x, ⋅)⟩ µ β ≤ C √ β ∑ x ∈ Z d ( C √ β ) dist ( x,B r ) ∥∇ u ( x, ⋅)∥ L ( µ β ) + C √ β ∥∇( u − u )∥ L ( B r ,µ β ) . A similar computation works for the term (2.12)-(ii). We decompose the sum over the diameter of the charges(2.15) ∑ q ∈Q ⟨∇ q u ⋅ a q ∇ q ( u − u )⟩ µ β = ∑ diam q ≤ r ⟨∇ q u ⋅ a q ∇ q ( u − u )⟩ µ β ·„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„¶ (2.15) −( i ) + ∞ ∑ n = r + ∑ diam q = n ⟨∇ q u ⋅ a q ∇ q ( u − u )⟩ µ β ·„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„¶ (2.15) −( ii ) . Note that since the function u − u is supported in the ball B r , we can restrict the sums to the charges whosesupport intersects the ball B r . The term (2.15)-(i) involving the charges of diameter smaller than r can beestimated by the Cauchy-Schwarz and Young’s inequalities. We have RRRRRRRRRRR ∑ diam q ≤ r ⟨∇ q u ⋅ a q ∇ q ( u − u )⟩ µ β RRRRRRRRRRR ≤ ∑ diam q ≤ r e − c √ β ∥ q ∥ ∥ n q ∥ L ∞ ⎛⎝ ∑ z ∈ supp n q ∥∇ u ( z, ⋅)∥ L ( µ β ) ⎞⎠ ⎛⎝ ∑ z ∈ supp n q ∥∇( u − u )( z, ⋅)∥ L ( µ β ) ⎞⎠≤ ∑ diam q ≤ r e − c √ βC q ∥ q ∥ ⎛⎝ ∑ z ∈ supp n q ∥∇ u ( z, ⋅)∥ L ( µ β ) ⎞⎠ + ∑ diam q ≤ r C q e − c √ β ∥ q ∥ ⎛⎝ ∑ z ∈ supp n q ∥∇( u − u )( z, ⋅)∥ L ( µ β ) ⎞⎠ ≤ ∑ diam q ≤ r e − c √ βC q ∥ q ∥ ∑ z ∈ supp n q ∥∇ u ( z, ⋅)∥ L ( µ β ) + ∑ diam q ≤ r C q e − c √ β ∥ q ∥ ∑ z ∈ supp n q ∥∇( u − u )( z, ⋅)∥ L ( µ β ) . We then absorb the terms ∥ n q ∥ L ∞ and ∣ supp n q ∣ into the exponential term e − c √ β ∥ q ∥ (by reducing the value ofthe constant c ) and use the estimate, for each z ∈ Z d , ∑ diam q ≤ r e − c √ β ∥ q ∥ { z ∈ supp n q } ≤ ⎧⎪⎪⎨⎪⎪⎩ z ∉ B r ,Ce − c √ β if z ∈ B r , where we recall that the sum is restricted to the charges q ∈ Q such that the support of q intersects the ball B r .We obtain the inequality(2.16) RRRRRRRRRRR ∑ diam q ≤ r ⟨∇ q u ⋅ a q ∇ q ( u − u )⟩ µ β RRRRRRRRRRR ≤ Ce − c √ β ∥∇( u − u )( z, ⋅)∥ L ( B r ,µ β ) + Ce − c √ β ∥∇ u ( z, ⋅)∥ L ( B r ,µ β ) , where we used that the function u − u is supported in the ball B r .The same computation can be used to estimate the term (2.15)-(ii). We obtain, for each integer n ∈ N ,(2.17) RRRRRRRRRRR ∑ diam q = n ⟨∇ q u ⋅ a q ∇ q ( u − u )⟩ µ β RRRRRRRRRRR ≤ Ce − c √ βn ∥∇ u ∥ L ( B r + n ,µ β ) + Ce − c √ βn ∥∇ ( u − u )∥ L ( B r ,µ β ) . Combining the identity (2.15) with the estimates (2.16) and (2.17), we deduce that(2.18) ∑ q ∈Q ⟨∇ q u ⋅ a q ∇ q ( u − u )⟩ µ β ≤ Ce − c √ β ∥∇( u − u )∥ L ( B r ,µ β ) + Ce − c √ β ∑ x ∈ Z d e − c √ β dist ( x,B r ) ∥∇ u ( x, ⋅)∥ L ( µ β ) . We now combine the identity (2.12) with the estimates (2.14), (2.18) to obtain the inequality(2.19) ∑ y ∈ Z d ∥ ∂ y ( u − u )∥ L ( B r ,µ β ) + β ∥∇ ( u − u )∥ L ( B r ,µ β ) ≤ C ( e − c √ β + β ) ∥∇ ( u − u )∥ L ( B r ,µ β ) + Cβ − ∑ x ∈ Z d ( e − c √ β dist ( x,B r ) + ( Cβ − ) dist ( x,B r ) ) ∥∇ u ( x, ⋅)∥ L ( µ β ) . We choose the inverse temperature β large enough so that the coefficient C ( e − c √ β + β − ) is smaller that β .With this choice, the first term in the right side of the inequality (2.19) can be absorbed in the left side of thesame inequality. We obtain the estimate(2.20) ∥∇ ( u − u )∥ L ( B r ,µ β ) ≤ Cβ ∑ x ∈ Z d ( e − c √ β dist ( x,B r ) + ( Cβ − ) dist ( x,B r ) ) ∥∇ u ( x, ⋅)∥ L ( µ β ) . The inequality (2.10) can then be deduced from the estimate (2.20) thanks to the three ingredients listedbelow: ● By choosing β large enough, the exponential term ( Cβ − ) dist ( x,B r ) is smaller than 1 for any point x ∈ Z d . This leads to the estimate in the ball B r ∑ x ∈ B r ( e − c √ β dist ( x,B r ) + ( Cβ − ) dist ( x,B r ) ) ∥∇ u ( x, ⋅)∥ L ( µ β ) ≤ ∑ x ∈ B r ∥∇ u ( x, ⋅)∥ L ( µ β ) = ∥∇ u ∥ L ( B r ,µ β ) . ● We have the estimate in the annulus B R ∖ B r ,(2.21) ∑ x ∈ B R ∖ B r ( e − c √ β dist ( x,B r ) + ( Cβ − ) dist ( x,B r ) ) ∥∇ u ( x, ⋅)∥ L ( µ β ) ≤ ( e − c √ βr + ( Cβ − ) r ) ∥∇ u ( x, ⋅)∥ L ( B R ∖ B r ,µ β ) . Using the assumption r ≥ ( ln R ) and choosing β large enough, one obtains the estimate ∑ x ∈ B R ∖ B r ( e − c √ β dist ( x,B r ) + ( Cβ − ) dist ( x,B r ) ) ∥∇ u ( x, ⋅)∥ L ( µ β ) ≤ e − cr ln β ∥∇ u ∥ L ( B R ∖ B r ,µ β ) ≤ e − c ( ln R ) ln β ∥∇ u ∥ L ( B R ,µ β ) . ● For each point x ∈ Z d ∖ B R , we have the estimate c ∣ x ∣ ≤ dist ( x, B r ) ≤ C ∣ x ∣ . This implies ∑ x ∈ B R ∖ B r ( e − c √ β dist ( x,B r ) + ( Cβ − ) dist ( x,B r ) ) ∥∇ u ( x, ⋅)∥ L ( µ β ) ≤ ∑ x ∈ B R ∖ B r e − c ( ln β )∣ x ∣ ∥∇ u ( x, ⋅)∥ L ( µ β ) . A combination of the inequality (2.20) with the three previous items completes the proof of the estimate (2.10).We complete Step 1 by proving that the estimate (2.10) implies the estimate (2.8). We consider theregularity exponent ε fixed at the beginning of the proof and the parameter δ ε provided by Proposition 2.2(associated to the exponent ε ). We let C ∶= C ( d ) < ∞ and c ∶= c ( d ) > X ∶= ( ln R ) and K ∶= Ce − c ln β ( ln R ) ∥∇ u ∥ L ( B R ,µ β ) + C ∑ x ∈ Z d e − c ( ln β )( R ∨∣ x ∣) ∥ u ( x, ⋅)∥ L ( µ β ) . An application of Proposition 2.2 shows the inequality: for any radius r ∈ [ X, R ] ,(2.22) ∥∇ u ∥ L ( B r ,µ β ) ≤ C ( Rr ) ε ∥∇ u ∥ L ( B R ,µ β ) + e − c ln β ( ln R ) ∥∇ u ∥ L ( B R ,µ β ) + C ∑ x ∈ Z d e − c ( ln β )( R ∨∣ x ∣) ∥ u ( x, ⋅)∥ L ( µ β ) . . NASH-ARONSON ESTIMATE AND REGULARITY THEORY FOR THE HEAT KERNEL P f We then note that the exponential term e − c ( ln β )( ln R ) decays faster than any power of R , so the second termon the right side of (2.22) can be bounded from above by the first term on the right side. This completes theproof of the inequality (2.8). Step 2.
We select r = ( ln R ) , apply the Caccioppoli inequality to estimate the right side of the inequal-ity (2.8) and use that the discrete gradient is a bounded operator to replace the term ∥∇ u ( x, ⋅)∥ L ( µ β ) by ∥ u ( x, ⋅)∥ L ( µ β ) . We obtain ∥∇ u ∥ L ( B ( ln R ) ,µ β ) ≤ C ( R ( ln R ) ) ε R ∥ u − ( u ) B R ∥ L ( B R ,µ β ) + C ∑ x ∈ Z d e − c ( ln β )( R ∨∣ x ∣) ∥ u ( x, ⋅)∥ L ( µ β ) . We apply the discrete L ∞ − L -estimate ∥∇ u ( )∥ L ( µ β ) ≤ ∥∇ u ∥ L ( B ( ln R ) ,µ β ) ≤ ( ln R ) d ∥∇ u ∥ L ( B ( ln R ) ,µ β ) . We then combine the two previous displays and the estimate ( ln R ) d ≤ CR ε to obtain the inequality (2.7).The proof of Proposition 2.4 is complete. (cid:3)
3. Nash-Aronson estimate and regularity theory for the heat kernel P f The main purpose of this section is to prove upper bounds on the heat kernel P f and on its spatialderivatives. We introduce the following definition. For each constant C >
0, we let Φ C be the function definedfrom ( , ∞) × Z d to R by the formula, for each pair ( t, x ) ∈ ( , ∞) × Z d ,(3.1) Φ C ( t, x ) = ⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩ t − d exp (− ∣ x ∣ Ct ) if ∣ x ∣ ≤ t, exp (− ∣ x ∣ C ) if ∣ x ∣ ≥ t. The next proposition is the main result of this section.
Proposition C , − ε -regularity for the heat kernel) . For any regularity exponent ε > , there exists an inverse temperature β ( d, ε ) < ∞ and a constant C ∶= C ( d, ε ) < ∞ such that for every β > β , every exponent p ∈ [ , ∞] and every random variable f ∈ L p ( µ β ) , the heat kernel P f satisfies thefollowing estimate, for each ( t, x, y ) ∈ ( , ∞) × Z d × Z d , (3.2) ∥P f ( t, x, ⋅ ; y )∥ L p ( µ β ) ≤ C ∥ f ∥ L p ( µ β ) Φ C ( tβ , x − y ) . Moreover, one has the C , − ε -estimates on the gradient of the heat kernel (3.3) ∥∇ x P f ( t, x, ⋅ ; y )∥ L p ( µ β ) ≤ C ∥ f ∥ L p ( µ β ) ( βt ) − ε Φ C ( tβ , x − y ) , and on the mixed derivative of the heat kernel (3.4) ∥∇ x ∇ y P f ( t, x, ⋅ ; y )∥ L p ( µ β ) ≤ C ∥ f ∥ L p ( µ β ) ( βt ) − ε Φ C ( tβ , x − y ) . Remark . The Nash-Aronson type estimate (3.3) is proved by Naddaf and Spencer in [ , Section2.2.2] in the case of the discrete Ginzburg-Landau interface model. Remark . Due to the discrete setting of the problem and the infinite range of the operator L , the heatkernel does not have Gaussian decay when the value ∣ x ∣ tends to infinity. Instead it decays exponentially fast;this justifies the introduction of the function Φ C . Remark . For later use, we need to keep track of the dependence of the constants in the inversetemperature β . Proof.
The first ingredient in the proof of Proposition 3.1 is the Feymann-Kac representation formulawhich is described at the beginning of Chapter 5 and recalled below. If we let ( φ t ) t ≥ be the diffusion processassociated to the Langevin dynamics (0.5), then one has the identity P f ( t, x, φ ; y ) = E φ [ f ( φ t ) P φ ⋅ ( t, x ; y )] , where P φ ⋅ (⋅ , ⋅ ; y ) is the solution of the parabolic system(3.5) ⎧⎪⎪⎨⎪⎪⎩ ∂ t P φ ⋅ (⋅ , ⋅ ; y ) + L φ t spat P φ ⋅ (⋅ , ⋅ ; y ) = ( , ∞) × Z d ,P φ ⋅ ( , ⋅ ; y ) = δ y in Z d , where L φ t spat denotes the time-dependent elliptic operator L φ t spat ∶= − β ∆ + β ∑ n ≥ β n (− ∆ ) n + + ∑ q ∈Q ∇ ∗ q ⋅ a q ( φ t )∇ q , The core of the argument is to prove the three following estimates on the heat kernel P φ ⋅ : there exists aconstant C ∶= C ( d, ε ) < ∞ such that for each realization of the diffusion process ( φ t ) t ≥ and each triplet ( t, x, y ) ∈ ( , ∞) × Z d × Z d ,(3.6) ⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩ ∣ P φ ⋅ ( t, x ; y )∣ ≤ C Φ C ( tβ , x − y ) , ∣∇ x P φ ⋅ ( t, x ; y )∣ ≤ C ( βt ) − ε Φ C ( tβ , x − y ) , ∣∇ x ∇ y P φ ⋅ ( t, x ; y )∣ ≤ C ( βt ) − ε Φ C ( tβ , x − y ) . The proof of these results is postponed to Propositions 3.5 and 3.7; we now show how to complete the proof ofProposition 3.1 assuming that the estimates (3.6) hold.Using that the Gibbs measure µ β is invariant under the Langevin dynamics (0.5), the inequality (3.2) is aconsequence of the estimates (3.6) and the following computation, for each triplet ( t, x, y ) ∈ ( , ∞) × Z d × Z d , ∥P f ( t, x, ⋅ ; y )∥ pL p ( µ β ) = E [∣P f ( t, x, φ ; y )∣ p ]= E [∣ E φ [ f ( φ t ) P φ ⋅ ( t, x, y )]∣ p ]≤ E [ E φ [∣ f ( φ t ) P φ ⋅ ( t, x, y )∣ p ]]≤ ( C Φ C ( tβ , x − y )) p E [ E φ [∣ f ( φ t )∣ p ]]≤ ( C Φ C ( tβ , x − y )) p ∥ f ∥ pL p ( µ β ) . The proofs of the estimates (3.3) and (3.4) is similar and we skip the details. (cid:3)
The rest of Section 3 is devoted to the statements and proofs of Propositions 3.5 and 3.7. Gaussian boundson the heat kernel are usually a consequence of the Nash-Aronson estimate (see [
8, 32 ]) for uniformly ellipticoperators. This result cannot be applied here since the operator ∂ t + L φ t spat is a parabolic system (see thecounter-example of De Giorgi [ ] disproving the Liouville property and the C ,α -regularity theory for systemsof elliptic equations).To prove Gaussian bounds and regularity on the heat kernel, we proceed differently and organize the proofas follows:(1) We use that the elliptic operator L φ t spat is a perturbation of the Laplacian to establish C , − ε -regularityfor the solutions of the system(3.7) ∂ t u + L φ t spat u = C , − ε -regularity and an interpolation argument to obtain L ∞ -bounds on the solutions ofthe equation (3.7). More precisely, we prove that every solution of the system (3.7) in the paraboliccylinder Q r satisfies the pointwise estimate ∥ u ∥ L ∞ ( Q r ) ≤ C ∥ u ∥ L ( Q r ) + ∫ − r ∑ x ∈ Z d ∖ B r e − c ( ln β )( r ∨∣ x ∣) ∣ u ( t, x )∣ dt ;(3) We prove that the solutions of the adjoint of the parabolic operator ∂ t + L φ t spat satisfies the samepointwise estimate; . NASH-ARONSON ESTIMATE AND REGULARITY THEORY FOR THE HEAT KERNEL P f (4) We use the pointwise regularity estimates and the technique Fabes and Stroock [ ], which is basedon the technique of Davies [
26, 27 ] (see also the article of Hofmann and Kim [ ] on which theargument is based) to establish the Gaussian bounds on the heat kernel stated in Proposition 3.5;(5) We combine the Gaussian bounds on the heat kernel with the C − ε -regularity theory for the solutionsof (3.7) to obtain the upper bounds on the gradient and mixed derivative of the heat kernel stated inProposition 3.7. This section is de-voted to the statement and proof of Proposition 3.5; as in Sections 1 and 2, the infinite range of the operator L φ t spat has to be taken into consideration in the analysis. Proposition . There exists an inverse temperature β ( d ) < ∞ suchthat for any point y ∈ Z d and any time-dependent continuous field φ ∶ R × Z d ↦ R , if we denote by P φ ⋅ (⋅ , ⋅ ; y ) the solution of the parabolic system ⎧⎪⎪⎨⎪⎪⎩ ∂ t P φ ⋅ (⋅ , ⋅ ; y ) + L φ t spat P φ ⋅ (⋅ , ⋅ ; y ) = in ( , ∞) × Z d ,P φ ⋅ ( , ⋅ ; y ) = δ y in Z d , then there exists a constant C ∶= C ( d ) < ∞ such that one has the estimate, for each pair ( t, x ) ∈ ( , ∞) × Z d , (3.8) ∣ P φ ⋅ ( t, x ; y )∣ ≤ C Φ C ( tβ , x − y ) . Remark . Assuming that the field φ is defined on the entire time line R is unnecessary; one couldassume that it is only define on the interval of positive times [ , ∞) . We make this assumption because it isconvenient in the argument and does not cause any loss of generality. Proof.
We first simplify the problem by removing some dependence of the parameters in the inversetemperature β . By the change of variable t → tβ , to prove the estimate (3.8), it is sufficient to prove that forevery continuous field φ ∶ R × Z d → R , the solution of the parabolic system(3.9) ⎧⎪⎪⎨⎪⎪⎩ ∂ t ̃ P φ ⋅ (⋅ , ⋅ ; y ) + β L φ t spat ̃ P φ ⋅ (⋅ , ⋅ ; y ) = ( , ∞) × Z d , ̃ P φ ⋅ ( , ⋅ ; y ) = δ y in Z d , satisfies the estimate(3.10) ∣ ̃ P φ ⋅ ( t, x ; y )∣ ≤ C Φ C ( t, x − y ) . We now prove the estimate (3.10) following the sketch of the argument described in Section 3. We fix atime-continuous field φ ∶ R × Z d → R . Step 1.
We first treat the point (1) and establish the C , − ε -regularity of the solutions of the equation (3.7).More precisely, we prove the following result: for each regularity exponent ε >
0, there exists an inversetemperature β ∶= β ( d, ε ) < ∞ and constants C ∶= C ( d, ε ) < ∞ and c ∶= c ( d ) > β ≥ β , each radius r >
1, each pair ( t, x ) ∈ R × Z d and each function u ∶ [− r + t, t ) × Z d → R ( d ) solution of the parabolic system(3.11) ∂ t u + β L φ t spat u = Q r ( t, x ) , one has the estimate(3.12) [ u ] C , − ε ( Q r ( t,x )) ≤ Cr − ε ∥ u − ( u ) Q r ∥ L ( Q r ( t,x )) + ∫ t − r + t ∑ y ∈ Z d e − c ( ln β )( r ∨∣ y − x ∣) ∣ u ( s, y )∣ ds. We follow the arguments of [ ] and assume without loss of generality that t = x =
0. We decompose theproof of (3.12) into three substeps: ● In Substep 1, we use that the operator L φ t spat is a perturbation of the Laplacian to prove that thefunction u is well-approximated by caloric functions. More specifically, we prove the following result:for each parameter δ >
0, there exists and inverse temperature β ( d, δ ) < ∞ such that for each β ≥ β each radius r >
1, there exists a function u caloric on the cylinder Q r such that one has the estimate(3.13) ∥∇( u − u )∥ L ( Q r ) ≤ δ ∥∇ u ∥ L ( Q r ) + ∫ − r ∑ x ∈ Z d e − c ( ln β )( r ∨∣ x ∣) ∣∇ u ( t, x )∣ dt. ● In Substep 2, we use the regularity known on the caloric functions to deduce from Step 1 that foreach regularity exponent ε >
0, there exists an inverse temperature β ∶= β ( d, ε ) < ∞ such that foreach β ≥ β and each pair of radii r, R ∈ ( , ∞) with r ≤ R ,(3.14) ∥ u − ( u ) Q r ∥ L ( Q r ) ≤ C ( rR ) − ε ∥ u − ( u ) Q R ∥ L ( Q R ) + ∫ − R ∑ x ∈ Z d e − c ( ln β )( R ∨∣ x ∣) ∣ u ( t, x )∣ dt. ● The C , − ε -regularity estimate (3.12) can be deduced from the estimate (3.14) by the integralcharacterization of H¨older continuous functions due to Meyers [ ]. The adaptation to the discretesetting being straightforward and we omit the details. Substep 1.
The proof can essentially be extracted from the first step of the proof of Proposition 2.4. Welet u be the solution of the parabolic boundary value problem ⎧⎪⎪⎪⎨⎪⎪⎪⎩ ∂ t u −
12 ∆ u = Q r ,u = u on ∂ ⊔ Q r , where the notation ∂ ⊔ Q r denotes the parabolic boundary of the cylinder ∂ ⊔ Q r (see Section 1.4 of Chapter 2).We then apply the proof of Step 1 of Proposition 2.4 to obtain the result. There are two differences in thedemonstration: we do not have a Laplacian in the φ -variable as in (2.9) and the problem is not elliptic butparabolic, nevertheless the extension of the proof Proposition 2.4 requires a mostly notational modification ofthe argument so we omit the details. Substep 2.
The strategy is similar to the one presented in Proposition 2.4 and follows standard arguments;we only give a sketch of the proof. We choose the inverse temperature β large enough so that the estimate (3.13)holds with the parameters δ ∶= δ ε , where δ ε is the parameter which appears in the statement of Lemma 2.3associated to the regularity exponent ε /
2. We apply Lemma 2.3 with the function φ ( r ) ∶= ∥∇ u ∥ L ( Q r ) to obtainthat, for each pair of radii r, R ≥ R ≥ r ≥ ( ln R ) ,(3.15) ∥∇ u ∥ L ( Q r ) ≤ C ( Rr ) ε ∥∇ u ∥ L ( Q R ) + ∫ − R ∑ x ∈ Z d ∖ B R e − c ( ln β )∣ x ∣ ∣∇ u ( t, x )∣ dt. We estimate the terms on the left and right side of the inequality thanks to the Caccioppoli inequality foruniformly elliptic parabolic equations and the Poincar´e inequality for solutions of parabolic equations (see forinstance [
71, 42 ]). These estimates should be adapted to the specific setting of the infinite range operatorconsidered here; this can be achieved by using the approximation arguments presented in Section 2. We obtainthe estimate(3.16) ∥ u − ( u ) Q r ∥ L ( Q r ) ≤ C ( rR ) − ε ∥ u − ( u ) Q R ∥ L ( Q R ) + ∫ − R ∑ x ∈ Z d e − c ( ln β )( R ∨∣ x ∣) ∣ u ( t, x )∣ dt. There remains to extend the inequality (3.16) to the small radii r ∈ [ , ( ln R ) ] . We use the inequality ∥ u − ( u ) Q r ∥ L ( Q r ) ≤ ⎛⎝ ∣ Q ( ln R ) ∣∣ Q r ∣ ⎞⎠ d ∥ u − ( u ) Q ( ln R ) ∥ L ( Q ( ln R ) ) ≤ C ( ln R ) d ( d + ) ∥ u − ( u ) Q ( ln R ) ∥ L ( Q ( ln R ) ) . We complete the argument by applying the inequality (3.16) to estimate the L ( Q ( ln R ) ) -norm of ∇ u andapply the estimate, valid under the assumption r ≤ ( ln R ) , ( ln R ) d ( d + ) ≤ C ( Rr ) ε . This completes the proof of Substep 2 and of the point (1).
Step 2.
We now treat the point (2); the objective is to deduce from the inequality (3.12) the pointwiseestimate(3.17) ∥ u ∥ L ∞ ( Q r ) ≤ C ∥ u ∥ L ( Q r ) + ∫ − r ∑ x ∈ Z d e − c ( ln β )( r ∨∣ x ∣) ∣ u ( t, x )∣ dt. . NASH-ARONSON ESTIMATE AND REGULARITY THEORY FOR THE HEAT KERNEL P f To prove the estimate (3.17), we interpolate the space L ∞ between the spaces L and C , − ε according to theformula, for any function u ∶ Q r → R , ∥ u ∥ L ∞ ( Q r ) ≤ C ∥ u ∥ αL ( Q r ) [ u ] − αC − ε ,Q r , with the explicit value α = ( − ε ) d + + ( − ε ) . Applying the estimate (3.12), we obtain ∥ u ∥ L ∞ ( Q r ) ≤ C ∥ u ∥ − αL ( Q r ) ⎛⎝∥ u ∥ L ( Q r ) + ∫ − r ∑ x ∈ Z d ∖ B r e − c ( ln β )( r ∨∣ x ∣) ∣ u ( t, x )∣ dt ⎞⎠ α ≤ C ∥ u ∥ L ( Q r ) + ∫ − r ∑ x ∈ Z d e − c ( ln β )( r ∨∣ x ∣) ∣ u ( t, x )∣ dt. The proof of (3.17) is complete.
Step 3.
We now treat the point (3). The adjoint system of (3.11) is given by ∂ t v − β L φ t spat v = , so that a formal integration by parts leads to the identity ∬ ( ∂ t + β L φ t spat ) u ⋅ v + ( ∂ t − β L φ t spat ) v ⋅ u = . For each point x ∈ Z d and each radius r ≥
1, we denote by Q ∗ r ( x ) the parabolic cylinder associated to the dualsystem Q ∗ r ∶= ( , r ) × B r . This operator ( ∂ t − β L φ t spat ) is a perturbation of ( ∂ t − ∆ ) . One can thus apply the same arguments as theones developed for the operator ( ∂ t + β L φ t spat ) to prove the L ∞ − L -regularity estimate, for each function v ∶ Q ∗ r → R , solution of the parabolic system ∥ v ∥ L ∞ ( Q ∗ r ) ≤ C ∥ v ∥ L ( Q ∗ r ) + C ∫ r ∑ x ∈ Z d e − c ( ln β )( r ∨∣ x ∣) ∣∇ v ( t, x )∣ dt. This completes the proof of the point (3).
Step 4.
We now treat the point (4). We fix a Lipschitz function ψ from Z d to R . We denote by γ theLipschitz constant of the function ψ and we always assume through the argument that γ ≤
1. We first recordfour inequalities. The first three estimates involve the discrete gradient of the function ψ . They read as follows:there exists a constant C ∶= C ( d ) < ∞ such that(3.18) ⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩ ∣∇ e ψ ∣ ≤ Cγe ψ , ∣∇ e − ψ ∣ ≤ Cγe − ψ ∀ A ⊆ Z d , sup A e ψ ≤ e diam A inf A e ψ , where we used γ ≤ ∇ and is stated below. Since the iterated gradient is an operator which has range n , one hasthe estimate, for each integer n ∈ N , each function v ∶ Z d → R ( d ) and each point x ∈ Z d ,(3.19) ⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩ ∣∇ n v ( t, x )∣ ≤ C n − ∑ y ∈ B ( x,n ) ∣∇ v ( t, x )∣ , ∣∇ n v ( x )∣ ≤ C n ∑ y ∈ B ( x,n ) ∣ v ( x )∣ , ∣∇ n ( e ψ v ) ( x ) − e ψ ( x ) ( x )∇ n v ( x )∣ ≤ C n γ ∑ y ∈ B ( x,n ) e ψ ( y ) ∣ v ( y )∣ . We now let K be a large constant whose value is decided at the end of the proof and should only dependon the dimension d . Given a time s ∈ R and a point y ∈ Z d , we let Γ (⋅ , ⋅ ; y, s ) ∶ ( , ∞) × Z d → R ( d )×( d ) be theparabolic Green’s matrix, i.e., the solution of the parabolic system(3.20) ⎧⎪⎪⎨⎪⎪⎩ ∂ t Γ (⋅ , ⋅ ; y, s ) + β L φ ⋅ spat Γ (⋅ , ⋅ ; y, s ) = ( s, ∞) × Z d , Γ ( s, ⋅ ; y, s ) = δ y in Z d . In particular we have the identity ̃ P φ ⋅ = Γ (⋅ , ⋅ ; ⋅ , ) . We introduce the notation Γ because it gives someadditional degrees of freedom regarding the starting time and point. We then let Q s → t be the operator actingon compactly supported functions f ∶ Z d → R ( d ) according to the formula, for each x ∈ Z d , Q ψs → t f ( x ) ∶= e − ψ ( x ) ∑ y ∈ Z d e ψ ( y ) Γ ( x, t ; y, s ) f ( y ) , and we note that the function v ( t, x ) ∶= e ψ ( x ) Q ψs → t f is solution of the parabolic system ∂ t v + β L φ ⋅ spat v =
0. Wecompute ∂ t ∥ Q ψs → t f ∥ L ( Z d ) = ∂ t ∑ x ∈ Z d ∣ e ψ ( x ) v ( t, x )∣ (3.21) = ∑ x ∈ Z d e ψ ( x ) v ( t, x ) ∂ t v ( t, x )= ∑ x ∈ Z d e ψ ( x ) v ( t, x )L φ ⋅ spat v ( t, x )= − ∑ x ∈ Z d ∇ ( e ψ ( x ) v ( t, x )) ∇ v ( t, x )·„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„¶ (3.21) −( i ) − ∑ n ≥ β n ∑ x ∈ Z d ∇ n + ( e ψ ( x ) v ( t, x )) ∇ n + v ( t, x )·„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„¶ (3.21) −( ii ) + β ∑ q ∈Q ∇ q ( e ψ ( x ) v ( t, x )) a q ∇ q v ( t, x )·„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„¶ (3.21) −( iii ) . We estimate the three terms (3.21)-(i), (3.21)-(ii), (3.21)-(iii) separately. For the term (3.21)-(i), we expandthe gradient of the product, use the inequality (3.18) and Young’s inequality. We obtain − ∑ x ∈ Z d ∇ ( e ψ ( x ) v ( t, x )) ∇ v ( t, x ) = − ∑ x ∈ Z d e ψ ( x ) ∣∇ v ( t, x )∣ − ∑ x ∈ Z d (∇ e ψ ) ( x ) v ( t, x )∇ v ( t, x ) (3.22) ≤ − ∑ x ∈ Z d e ψ ( x ) ∣∇ v ( t, x )∣ + Cγ ∑ x ∈ Z d e ψ ( x ) ∣ v ( t, x )∣ ∣∇ v ( t, x )∣≤ − ∑ x ∈ Z d e ψ ( x ) ∣∇ v ( t, x )∣ + Cγ ∑ x ∈ Z d e ψ ( x ) ∣ v ( t, x )∣ ≤ − ∑ x ∈ Z d e ψ ( x ) ∣∇ v ( t, x )∣ + Cγ ∥ Q ψs → t f ∥ L ( Z d ) . For the term (3.21)-(ii), we use the inequality (3.19) and obtain(3.23) ∑ x ∈ Z d ∇ n ( e ψ v ( t, ⋅)) ( x )∇ n v ( t, x ) ≤ ∑ x ∈ Z d e ψ ( x ) ∣∇ n v ( t, x )∣ + C n γ ∑ x ∈ Z d ∑ y ∈ B ( x,n ) e ψ ( y ) ∣ v ( t, y )∣ ∣∇ n v ( t, x )∣ . We use the inequalities (3.19) a second time, the property (3.18) and Young’s inequality. We obtain(3.24) ∑ x ∈ Z d ∇ n ( e ψ v ( t, ⋅)) ( x )∇ n v ( t, x ) ≤ C n ∑ x ∈ Z d e ψ ( x ) ∣∇ v ( t, x )∣ + C n γ ∑ x ∈ Z d e ψ ( x ) ∣ v ( t, x )∣ . We then multiply the inequality (3.24) by β − n and sum over the integers n ∈ N . We obtain ∑ n ≥ β n ∑ x ∈ Z d ∇ n + ( e ψ v ( t, ⋅)) ( x )∇ n + v ( t, x )≤ ( ∑ n ≥ ( C √ β ) n ) ∑ x ∈ Z d e ψ ( x ) ∣∇ v ( t, x )∣ + ( ∑ n ≥ ( C √ β ) n ) γ ∑ x ∈ Z d e ψ ( x ) ∣ v ( t, x )∣ . By choosing β larger than the square of the constant C , we deduce(3.25) ∑ n ≥ β n ∑ x ∈ Z d ∇ n + ( e ψ v ( t, ⋅)) ( x )∇ n v ( t, x ) ≤ Cβ ∑ x ∈ Z d e ψ ( x ) ∣∇ v ( t, x )∣ + Cγ β ∑ x ∈ Z d e ψ ( x ) ∣ v ( t, x )∣ . . NASH-ARONSON ESTIMATE AND REGULARITY THEORY FOR THE HEAT KERNEL P f For the term (3.21)-(iii), we fix a charge q ∈ Q . We recall the bound ∣ a q ∣ ≤ e − c √ β ∥ q ∥ , the conventional notationfor the constant C q and the estimates (2.6) of Chapter 2. We compute ∇ q ( e ψ v ( t, ⋅)) a q ∇ q v ( t, ⋅) = ( d ∗ ( e ψ v ( t, ⋅)) , n q ) a q ∇ q v ( t, ⋅)≤ C q e − c √ β ∥ q ∥ ∥∇ ( e ψ v ( t, ⋅))∥ L ( supp n q ) ∥∇ v ( t, ⋅)∥ L ( supp n q ) . We then expand the gradient and use the properties (3.18) ∇ q ( e ψ v ( t, ⋅)) a q ∇ q v ( t, ⋅) ≤ C q e − c √ β ∥ q ∥ ∥∇ ( e ψ ) v ( t, ⋅)∥ L ( supp n q ) ∥∇ v ( t, ⋅)∥ L ( supp n q ) + C q e − c √ β ∥ q ∥ ∥ e ψ ∇ v ( t, ⋅)∥ L ( supp n q ) ∥∇ v ( t, ⋅)∥ L ( supp n q ) ≤ C q e − c √ β ∥ q ∥ γ ( sup supp n q e ψ ) ∥ e ψ v ( t, ⋅)∥ L ( supp n q ) ∥∇ v ( t, ⋅)∥ L ( supp n q ) + C q e − c √ β ∥ q ∥ ⎛⎝ sup supp nq e ψ ⎞⎠ ∥ e ψ ∇ v ( t, ⋅)∥ L ( supp n q ) ∥∇ v ( t, ⋅)∥ L ( supp n q ) . We use the property (3.18) (with the set A = supp n q ). We obtain ∇ q ( e ψ v ( t, ⋅)) a q ∇ q v ( t, ⋅) ≤ C q C diam n q e − c √ β ∥ q ∥ γ ∥ e ψ v ( t, ⋅)∥ L ( supp n q ) ∥ e ψ ∇ v ( t, ⋅)∥ L ( supp n q ) + C q C diam n q e − c √ β ∥ q ∥ ∥ e ψ ∇ v ( t, ⋅)∥ L ( supp n q ) ∥ e ψ ∇ v ( t, ⋅)∥ L ( supp n q ) . We choose the inverse temperature β large enough (depending only on the dimension d ) so that the constants C q and C diam n q can be absorbed by the exponential term e − c √ β ∥ q ∥ and apply the Young’s inequality. Weobtain ∇ q ( e ψ v ( t, ⋅)) a q ∇ q v ( t, ⋅) ≤ Ce − c √ β ∥ q ∥ γ ∥ e ψ v ( t, ⋅)∥ L ( Z d ) + Ce − c √ β ∥ q ∥ ∥ e ψ ∇ v ( t, ⋅)∥ L ( Z d ) . Summing over all the charges q ∈ Q and using the inequality, for each point x ∈ Z d , ∑ x ∈ Z d e − c √ β ∥ q ∥ { x ∈ supp n q } ≤ Ce − c √ β , we obtain the estimate β ∑ q ∈Q ∇ q ( e ψ v ( t, ⋅)) a q ∇ q v ( t, ⋅) ≤ Cβe − c √ β γ ∥ e ψ v ( t, ⋅)∥ L ( Z d ) + Cβe − c √ β ∥ e ψ ∇ v ( t, ⋅)∥ L ( Z d ) (3.26) ≤ Ce − c √ β γ ∥ e ψ v ( t, ⋅)∥ L ( Z d ) + Ce − c √ β ∥ e ψ ∇ v ( t, ⋅)∥ L ( Z d ) , where we have absorbed the coefficient β into the exponential terms e − c √ β in the second line. We combinethe estimates (3.21), (3.22), (3.25), (3.26) and choose β large enough so that the term Cβ − in the right sideof (3.25) and the term Ce − c √ β in the right side of (3.26) are both smaller than . We obtain the estimate ∂ t ∥ Q ψs → t f ∥ L ( Z d ) ≤ − ∑ x ∈ Z d e ψ ( x ) ∣∇ v ( t, x )∣ + Cγ ∥ Q ψs → t f ∥ L ( Z d ) (3.27) ≤ Cγ ∥ Q ψs → t f ∥ L ( Z d ) . By integrating the equation (3.27) between the times s and t , we obtain the inequality ∥ Q ψs → t f ∥ L ( Z d ) ≤ e Cγ ( t − s ) ∥ f ∥ L ( Z d ) . The adjoint of the operator Q s → t is given by the formula, for each compactly supported function g ∶ Z d → R ( d ) , ( Q ψs → t ) ∗ g ( y ) = e − ψ ( y ) ∑ x ∈ Z d e ψ ( x ) Γ ∗ ( y, s ; x, t ) g ( x ) , where Γ ∗ ( x, t ; y, s ) is the fundamental solution of the dual operator ∂ t − β L φ ⋅ spat . By similar computation, weobtain the estimate ∥( Q ψs → t ) ∗ g ∥ L ( Z d ) ≤ e Cγ ( t − s ) ∥ g ∥ L ( Z d ) . Considering the specific function ψ = γ = L -estimates ∥ Q s → t f ∥ L ( Z d ) ≤ ∥ f ∥ L ( Z d ) and ∥( Q s → t ) ∗ g ∥ L ( Z d ) ≤ ∥ g ∥ L ( Z d ) . We then set u ( t, x ) = Q s → t f , use the L ∞ − L regularity estimate (3.17) in the parabolic cylinder Q √ t − s ( t, x ) and the boundedness of the discrete gradient in L ( Z d ) . We obtain ∣ u ( t, x )∣ ≤ C ( t − s ) + d ∫ ts ∑ y ∈ B ( x, √ t − s ) ∣ u ( t ′ , y )∣ + ∑ y ∈ Z d e − c ( ln β )(√ t − s ∨∣ y − x ∣) ∣ u ( t ′ , y )∣ dt ′ ≤ C ( t − s ) + d ∫ ts ∥ u ( t ′ , ⋅)∥ L ( Z d ) dt ′ ≤ C ( t − s ) d ∥ f ∥ L ( Z d ) . Since the previous inequality is valid for any point x ∈ Z d , we have obtained the following L ∞ − L -inequality ∥ Q s → t f ∥ L ∞ ( Z d ) ≤ C ( t − s ) d ∥ f ∥ L ( Z d ) . With the same computation, we obtain the L ∞ − L -estimate for the dual operator ( Q s → t ) ∗ (3.28) ∥( Q s → t ) ∗ g ∥ L ∞ ( Z d ) ≤ C ( t − s ) d ∥ g ∥ L ( Z d ) . In the general case, we apply the regularity estimate (3.17) with the function e ψ ( x ) Q ψs → t f and the radius r = √ t − s . We obtain(3.29) e ψ ( x ) ∣ Q ψs → t f ( t, x )∣ ≤ C ( t − s ) + d ∫ ts ∑ y ∈ B ( x, √ t − s ) e ψ ( y ) ∣ Q ψs → t f ( y )∣ + ∑ y ∈ Z d e − c ( ln β )(√ t − s ∨∣ y − x ∣) e ψ ( y ) ∣ Q ψs → t f ( y )∣ dt ′ . We then multiply each side of the inequality (3.29) by e − ψ ( x ) and note that we have the estimate, for each y ∈ Z d , e − ψ ( x ) e ψ ( y ) ≤ exp ( γ ∣ x − y ∣) . We obtain the estimate(3.30) ∣ Q ψs → t f ( x )∣ ≤ C ( t − s ) + d ∫ ts ∑ y ∈ B ( x, √ t − s ) e γ ∣ x − y ∣ ∣ Q ψs → t f ( y )∣ + ∑ y ∈ Z d e − c ( ln β )(√ t − s ∨∣ y − x ∣)+ γ ∣ y − x ∣ ∣ Q ψs → t f ( y )∣ dt ′ . We assume that the inverse temperature β is chosen large enough so that c ln β ≥
2, where c is the constantwhich appears in the exponential term e − c ( ln β )(√ t − s ∨∣ y − x ∣) in the right side of the inequality (3.30). Using thisassumption γ ≤
1, we obtain the estimate ∣ Q ψs → t f ( x )∣ ≤ C ( t − s ) + d ∫ ts ∑ y ∈ B ( x, √ t − s ) e γ √ t − s ∣ Q ψs → t f ( y )∣ + ∑ y ∈ Z d ∖ B ( x, √ t − s ) ∣ Q ψs → t f ( y )∣ dt ′ (3.31) ≤ Ce γ √ s − t ( t − s ) + d ∫ ts ∥ Q ψs → t ′ f ∥ L ( Z d ) dt ′ ≤ Ce γ √ s − t ( t − s ) + d ∫ ts e Cγ ( t ′ − s ) ∥ f ∥ L ( Z d ) dt ′ ≤ Ce γ √ s − t γ ( t − s ) + d e Cγ ( t − s ) ∥ f ∥ L ( Z d ) . . NASH-ARONSON ESTIMATE AND REGULARITY THEORY FOR THE HEAT KERNEL P f Since the previous estimate is valid for any point x ∈ Z d , we have obtained the following L ∞ − L estimate forthe operator Q ψs → t ,(3.32) ∥ Q ψs → t f ∥ L ∞ ( Z d ) ≤ Cγ ( t − s ) + d exp ( γ √ s − t + Cγ ( t − s )) ∥ f ∥ L ( Z d ) . A similar argument applies for the dual operator ( Q ψs → t ) ∗ and we obtain(3.33) ∥( Q ψs → t ) ∗ g ∥ L ∞ ( Z d ) ≤ Cβγ ( t − s ) + d exp ( γ √ s − t + Cγ ( t − s )) ∥ g ∥ L ( Z d ) . By duality the estimates (3.28) and (3.32) implies the inequalities ∥( Q s → t ) f ∥ L ( Z d ) ≤ C ( t − s ) d ∥ f ∥ L ( Z d ) , ∥( Q ψs → t ) f ∥ L ( Z d ) ≤ Cγ ( t − s ) + d exp ( γ √ s − t + Cγ ( t − s )) ∥ f ∥ L ( Z d ) . We then set τ = t + s and use the semigroup property Q ψs → t = Q ψs → τ ○ Q ψτ → t . We obtain the estimate(3.34) ∥( Q s → t ) f ∥ L ∞ ( Z d ) ≤ C ( t − s ) d ∥ f ∥ L ( Z d ) , ∥( Q ψs → t ) f ∥ L ∞ ( Z d ) ≤ Cγ ( t − s ) + d exp ( γ √ s − t + Cγ ( t − s )) ∥ f ∥ L ( Z d ) . The estimate (3.34) implies the upper bounds on the Green’s matrix Γ ( t, x ; s, y ) (3.35) ∣ Γ ( t, x ; s, y )∣ ≤ C ( t − s ) d ,e ψ ( x )− ψ ( y ) ∣ Γ ( t, x ; s, y )∣ ≤ C exp ( γ √( t − s ) + Cγ ( t − s )) γ ( t − s ) + d . We choose the function ψ according to the formula ψ ( z ) = γ ∣ z − y ∣ . The estimate (3.35) becomes ∣ Γ ( t, x ; s, y )∣ ≤ C ( t − s ) d min ⎛⎜⎝ , exp ( γ √( t − s ) + C γ ( t − s ) − γ ∣ x − y ∣) γ ( t − s ) ⎞⎟⎠ . We select the value of the coefficient γ : we let C ∶= C ( d ) < ∞ be a large constant and set γ ∶= ⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩ ∣ x − y ∣ C ( t − s ) if ∣ x − y ∣ ≤ ( t − s ) ,γ = C if ∣ x − y ∣ ≥ ( t − s ) . By choosing the constant C large enough depending on the dimension d , we obtain the following result. Thereexists a constant C ∶= C ( d ) < ∞ such that ∣ Γ ( t, x ; s, y )∣ ≤ ⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩ C ( t − s ) d exp (− ∣ x − y ∣ C ( t − s ) ) if ∣ x − y ∣ ≤ t − s,C exp (− ∣ x − y ∣ C ) if ∣ x − y ∣ ≥ t − s. Choosing the specific values y = s = (cid:3) This section is devoted tothe proof of Proposition 3.7 which establishes pointwise bounds on the gradient and mixed derivative of theheat kernel in terms of the regularity exponent ε and the function Φ C . Proposition C , − ε -regularity for the heat kernel) . For each exponent ε > , there exists an inversetemperature β ∶= β ( d, ε ) < ∞ such that the following result holds. For any β ≥ β , there exists a constant C ∶= C ( d, ε ) < ∞ such that for each triplet ( t, x, y ) ∈ ( , ∞) × Z d × Z d and any time-continuous coefficient field φ ∶ R × Z d → R , one has the estimate (3.36) ∣∇ x P φ ⋅ ( t, x ; y )∣ ≤ C ( βt ) + d − ε Φ C ( tβ , x − y ) , and (3.37) ∣∇ x ∇ y P φ ⋅ ( t, x ; y )∣ ≤ C ( βt ) + d − ε Φ C ( tβ , x − y ) . Proof.
By performing the change of variable t → tβ , it is sufficient to prove that the heat kernel ̃ P φ ⋅ defined in (3.9) satisfies the estimates ∣∇ x ̃ P φ ⋅ ( t, x ; y )∣ ≤ Ct + d − ε Φ C ( t, x − y ) and ∣∇ x ∇ y ̃ P φ ⋅ ( t, x ; y )∣ ≤ Ct + d − ε Φ C ( t, x − y ) . We fix an exponent ε >
0, a time t ≥ x ∈ Z d . We let β ∶= β ( d, ε ) < ∞ be the inversetemperature above which the the Gaussian bound (3.10) and the regularity estimate (3.12) are valid. Weapply the inequality (3.12) with the function u = ̃ P φ ⋅ (⋅ , ⋅ ; y ) in the parabolic cylinder Q √ t ( t, x ) and obtain theestimate(3.38) [ ̃ P φ ⋅ ] C , − ε ⎛⎝ Q √ t ( t,x )⎞⎠ ≤ Ct − ε XXXXXXXXXXXX ̃ P φ ⋅ − ( ̃ P φ ⋅ ) Q √ t ( t,x ) XXXXXXXXXXXX L ⎛⎝ Q √ t ( t,x )⎞⎠ + ∫ t t ∑ z ∈ Z d e − c ( ln β )(√ t ∨∣ z ∣) ∣ ̃ P φ ⋅ ( t ′ , z ; y )∣ dt ′ . We then use the Gaussian bound (3.10) to estimate the right side of (3.38). We obtain(3.39) [ ̃ P φ ⋅ ] C , − ε ⎛⎝ Q √ t ( t,x )⎞⎠ ≤ Ct − ε Φ C ( t, x − y ) . In the discrete setting, assuming that t ≥
8, we can write(3.40) ∣∇ ̃ P φ ⋅ ( t, x )∣ ≤ ∑ y ∼ x ∣ ̃ P φ ⋅ ( t, y ) − ̃ P φ ⋅ ( t, x )∣ ≤ C osc Q ( t,x ) ̃ P φ ⋅ ≤ C [ ̃ P φ ⋅ ] C , − ε ⎛⎝ Q √ t ( t,x )⎞⎠ . A combination of the estimates (3.39) and (3.40) completes the proof of the estimate (3.36). We note that thesame argument gives the more general bound involving the parabolic Green’s matrix Γ defined in (3.20): foreach pair of times 0 ≤ s < t < ∞ and each pair of points x, y ∈ Z d ,(3.41) ∣∇ x Γ ( t, x ; s, y )∣ ≤ C ( t − s ) − ε Φ C ( t − s, x − y ) . To prove the estimate (3.37) we use that the function ( s, y ) ↦ ∇ x Γ ( t, x ; s, y ) is solution of the dual equation ∂ s ∇ x Γ ( t, x ; ⋅ , ⋅) − β L φ ⋅ spat ,y ∇ x Γ ( t, x ; ⋅ , ⋅) =
0. We can thus apply the estimate (3.12) (since this C , − ε -regularityestimate also holds for the dual parabolic problem by the same perturbation argument) and the argumentsused in the proof of the inequality (3.36). We obtain ∣∇ x ∇ y Γ ( t, x ; s, y )∣ ≤ Ct − ε ∥∇ x Γ ( t, x ; ⋅ , ⋅)∥ L ⎛⎜⎝ Q ∗√ t − s ( s,y )⎞⎟⎠ + ∫ s + t s ∑ z ∈ Z d e − c ( ln β )(√ t − s ∨∣ z ∣) ∣∇ x Γ ( t, x ; t ′ , z )∣ dt ′ . We use the estimate (3.41) on the gradient of the function Γ to obtain(3.42) ∣∇ x ∇ y Γ ( t, x ; s, y )∣ ≤ C ( t − s ) − ε Φ C ( t − s, x − y ) . . REGULARITY ESTIMATE FOR THE DIFFERENTIATED HELFFER-SJ ¨OSTRAND EQUATION 83 There is an exponent 2 ε instead of ε in the right side of (3.42). Since this estimate is valid for any exponent ε > β if necessary), one can fix this issue by rewriting the proofwith the exponent ε instead of ε to obtain the desired upper bound. Finally the estimate (3.42) implies theinequality (3.37) by setting s = (cid:3) G f . . In this section,we consider the elliptic Green’s matrix associated to the operator L defined in (4.16) in Chapter 4. We fix anexponent p ∈ [ , ∞] , and a function f ∈ L p ( µ β ) . We consider the map P f introduced in Section 3 and definethe elliptic Green’s matrix G f ∶ Z d × Ω × Z d ↦ R by the formula, for each ( x, φ, y ) ∈ Z d × Ω × Z d ,(3.43) G f ( x, φ ; y ) ∶= ∫ ∞ P f ( t, x, φ ; y ) dt. As a consequence of the Feynman-Kac formula, this function can be equivalently characterized as the uniquesolution of the equation (see (4.16) in Chapter 4). LG f ( x, φ ; y ) = f ( φ ) δ y ( x ) in Z d × Ω , such that ∥G f ( x, ⋅ ; y )∥ L p ( µ β ) tends to 0 as x tends to infinity.Using the equivalent characterization by Feynman-Kac, and bounds on the mixed gradients of the heatkernel established in the previous section, we obtain asymptotic estimates on the L p ( µ β ) -norm of the Green’smatrix, its gradient and its mixed derivative. This proves Proposition 4.7 of Chapter 4. Proof of Proposition 4.7 of Chapter 4.
The proof is obtained by using the formula (3.43) andintegrating the estimates (3.2), (3.3) and (3.4) on the heat kernel over time. (cid:3)
4. Regularity estimate for the differentiated Helffer-Sj¨ostrand equation
In this section, we study the differentiated Helffer-Sj¨ostrand obtained by the following procedure. Weconsider a function u ∈ C ∞ c ( Z d × Ω ) and we denote by G ∶= L u . We apply the operator ∂ x to both the left andright hand sides of the identity L u = G . We obtain the identity(4.1) ∂ x ∆ φ u − β ∆ ∂ x u + β ∑ n ≥ β n (− ∆ ) n + ∂ x u + ∂ x ⎛⎝ ∑ q ∈Q ∇ ∗ q ⋅ a q ∇ q u ⎞⎠ = ∂ x G. To go further in the computation, we introduce the following notations: ● We define the function v, h ∶ Z d × Z d × Ω ↦ R ( d )×( d ) by the formulas, for each for ( x, y, φ ) ∈ Z d × Z d × Ω, v ( x, y, φ ) = ∂ x u ( y, φ ) and h ( x, y, φ ) = ∂ x G ( y, φ ) . ● Given a map h ∶ Z d × Z d × Ω ↦ R ( d )×( d ) , we denote by ∆ x the spatial Laplacian in the first variableand by d ∗ y the Laplacian in the second variable. We also denote by ∑ q x ∈Q ∇ ∗ q x ⋅ a q x ∇ q x h and by ∑ q y ∈Q ∇ ∗ q y ⋅ a q y ∇ q y h the operators ∑ q x ∈Q ∇ ∗ q x ⋅ a q x ∇ q x h ∶ ( x, y, φ ) ↦ ∑ q ∈Q a q ( φ ) ( h (⋅ , y, φ ) , q ) q ( x ) and ∑ q y ∈Q ∇ ∗ q y ⋅ a q y ∇ q y h ∶ ( x, y, φ ) ↦ ∑ q ∈Q a q ( φ ) ( h ( x, ⋅ , φ ) , q ) q ( y ) . ● Finally, we denote by L spat ,x and L spat ,y the operators L spat ,x ∶= − β ∆ x u + β ∑ n ≥ β n (− ∆ x ) n + u + ∑ q x ∈Q ∇ ∗ q x ⋅ a q x ∇ q x u, and L spat ,y ∶= − β ∆ y u + β ∑ n ≥ β n (− ∆ y ) n + u + ∑ q y ∈Q ∇ ∗ q y ⋅ a q y ∇ q y u, The term ∂ ⋅ ∆ φ u can be computed by using the same strategy as the one used to derive the Helffer-Sj¨ostrandequation in Section 4 of Chapter 3 and we obtain, for each ( x, y, φ ) ∈ Z d × Z d × Ω, ∂ x ∆ φ u ( y, φ ) = ∆ φ v ( x, y, φ ) − β ∆ x v ( x, y, φ ) + β ∑ n ≥ β n (− ∆ x ) n + v ( x, y, φ ) + ∑ q x ∈Q ∇ ∗ q x ⋅ a q x ∇ q x v ( x, y, φ ) (4.2) = ∆ φ v ( x, y, φ ) + L spat ,x v ( x, y, φ ) The term ∂ x (∑ q ∈Q ∇ ∗ q ⋅ a q ∇ q u ) can be computed by the exact formula stated in (0.2): ∂ x ⎛⎝ ∑ q ∈Q ∇ ∗ q ⋅ a q ∇ q u ( y, φ )⎞⎠ = ∂ x ⎛⎝ ∑ q ∈Q a q ( u, q ) q ( y )⎞⎠ (4.3) = ∑ q ∈Q ∂ x a q ( u, q ) q ( y ) + ∑ q ∈Q a q ( v ( x, ⋅ , φ ) , q ) q = ∑ q ∈Q πz ( β, q ) cos 2 π ( φ, q ) ( u, q ) q ( x ) ⊗ q ( y ) + ∑ q y ∈Q ∇ ∗ q y a q y ∇ q y v ( x, y, φ ) . Combining the identities (4.1), (4.2) and (4.3), we obtain that the map v solves the equation(4.4)∆ φ v ( x, y, φ ) + L spat ,x v ( x, y, φ ) + L spat ,y v ( x, y, φ ) = − ∑ q ∈Q πz ( β, q ) cos 2 π ( φ, q ) ( u, q ) q ( x ) q ( y ) + ∂ x G ( y, φ ) . This equality can be rigorously justified using the arguments of Section 4 of Chapter 3 and [
65, 40 ], we omitthe details. The identity (4.4) motivates the definition of the differentiated Helffer-Sj¨ostrand operator(4.5) L der ∶= ∆ φ + L spat ,x + L spat ,y . It is natural to consider the Green’s function associated to the differentiated Helffer-Sj¨ostrand equation(4.6) L der G der , f = f δ ( x,y ) in Z d × Z d × Ω . Notice that we can solve (4.6) variationally, by applying the Gagliardo-Nirenberg-Sobolev inequality as inLemma 4.6 of Chapter 3. P der , f . As in Sections 3and 3.3, we wish to study the properties of the heat kernel and the Green’s matrix associated to thedifferentiated Helffer-Sj¨ostrand operator. Given a real number p ∈ [ , ∞) , a function f ∈ L p ( µ β ) and apoint ( x, x ) ∈ Z d × Z d , we denote by P der , f (⋅ , ⋅ , ⋅ , ⋅ ; y, y ) ∶ ( , ∞) × Z d × Z d × Ω → R ( d ) the solution of the2 d -dimensional parabolic system ⎧⎪⎪⎨⎪⎪⎩ ∂ t P der , f (⋅ , ⋅ , ⋅ , ⋅ ; y, y ) + L der P der , f (⋅ , ⋅ , ⋅ , ⋅ ; y, y ) = ( , ∞) × Z d × Z d × Ω , P der , f ( , ⋅ , ⋅ , ⋅ ; y, y ) = f δ ( y,y ) in Z d × Z d × Ω . We then define the elliptic Green’s matrix G der , f according to the Duhamel principle by the formula, for each ( x, x , y, y ) ∈ ( Z d ) and each field φ ∈ Ω, G der , f ( x, x , φ ; y, y ) ∶= ∫ ∞ P der , f ( t, x, x , φ ; y, y ) dt. We denote by ∇ x , ∇ y , ∇ x , ∇ y the gradient with respect to the first, second, third and fourth spatial variablesof the maps P der , f and G der , f .We will prove in Proposition 4.1 below upper bounds on the heat kernel P der , f and its derivatives. Wethen combine this with the Duhamel principle to deduce upper bounds on the elliptic Green’s matrix G der , f inCorollary 4.2.Before stating the propositions, we make a few remarks about the results. If we let ( φ t ) t ≥ be the diffusionprocess defined by the formula (0.5) and if we recall the notation E φ introduced in the paragraph following (0.5),then one has the Feynman-Kac formula(4.7) P der , f ( t, x, x , φ ; y, y ) = E φ [ f ( φ t ) P φ ⋅ der ( t, x, x ; y, y )] , . REGULARITY ESTIMATE FOR THE DIFFERENTIATED HELFFER-SJ ¨OSTRAND EQUATION 85 where P φ ⋅ der (⋅ , ⋅ ; y ) is the solution of the system of equations,(4.8) ⎧⎪⎪⎪⎨⎪⎪⎪⎩ ∂ t P φ ⋅ der (⋅ , ⋅ , ⋅ ; y, y ) + (L φ t spat ,x + L φ t spat ,y ) P φ ⋅ der (⋅ , ⋅ , ⋅ ; y, y ) = ( , ∞) × Z d × Z d ,P φ ⋅ der ( , ⋅ , ⋅ ; y, y ) = δ ( y,y ) in Z d × Z d . The operator L φ ⋅ spat ,x + L φ ⋅ spat ,y is a uniformly elliptic operator on the 2 d -dimensional space Z d × Z d .Additionally, if the inverse temperature β is chosen large enough, then this operator is a perturbation of the2 d -dimensional Laplacian ∆ x + ∆ y . Hence the same arguments as in Section 3 can be used to prove Gaussianbounds and C , − ε -regularity estimates on the heat kernel P der , f ; the only difference is that the underlyingspace is 2 d -dimensional.The result stated in Proposition 4.1 is strictly stronger than the ones stated in Propositions 3.5 andProposition 3.7 since we obtain estimates on the triple and quadruple gradients of the heat kernel P der , f . Thisresults are obtained by making use of the specific structure of the problem as we now describe. The ellipticoperators L spat ,x and L spat ,y only acts on the x and y variables respectively; in particular they commute . Thisremark implies that heat kernel P φ ⋅ der defined in (4.8) factorises, i.e., we have the identity(4.9) P φ ⋅ der ( t, x, x ; y, y ) = P φ ⋅ ( t, x ; y ) ⊗ P φ ⋅ ( t, x ; y ) , where P φ ⋅ denotes the d -dimensional heat kernel defined in (3.5). Thanks to this property, one can obtainadditional regularity; for instance applying the gradruple gradient ∇ x ∇ x ∇ y ∇ y to the heat kernel gives ∇ x ∇ y ∇ x ∇ y P φ ⋅ der ( x, y ; x , y ) = ∇ x ∇ y P φ ⋅ ( t, x ; y ) ⊗ ∇ x ∇ y P φ ⋅ ( t, x ; y ) . We can then apply the regularity estimate (3.37) proved in Proposition 3.7.
Proposition . For any regularity exponent ε > , there exists an inverse temperature β ( d, ε ) < ∞ suchthat the following statement holds. For any inverse temperature β > β , there exists a constant C ( d, ε ) < ∞ such that for each ( x, y, x , y ) ∈ ( Z d ) , one has the estimate ∥P der , f ( t, x, x , ⋅ ; y, y )∥ L p ( µ β ) ≤ C ∥ f ∥ L p ( µ β ) Φ C ( tβ , x − x ) Φ C ( tβ , y − y ) , and the C , − ε -regularity estimates: if we let ∇ , ∇ , ∇ and ∇ be any permutation of the set of gradients ∇ x , ∇ x , ∇ y and ∇ y , then one has the four inequalities (i) On the gradient of the heat kernel ∥∇ P der , f ( t, x, x , ⋅ ; y, y )∥ L p ( µ β ) ≤ C ∥ f ∥ L p ( µ β ) ( βt ) − ε Φ C ( tβ , x − x ) Φ C ( tβ , y − y ) ;(ii) On the double gradient of the heat kernel ∥∇ ∇ P der , f ( t, x, x , ⋅ ; y, y )∥ L p ( µ β ) ≤ C ∥ f ∥ L p ( µ β ) ( βt ) − ε Φ C ( tβ , x − x ) Φ C ( tβ , y − y ) ;(iii) On the triple gradient of the heat kernel ∥∇ ∇ ∇ P der , f ( t, x, x , ⋅ ; y, y )∥ L p ( µ β ) ≤ C ∥ f ∥ L p ( µ β ) ( βt ) − ε Φ C ( tβ , x − x ) Φ C ( tβ , y − y ) ;(iv) On the quadruple gradient of the heat kernel ∥∇ ∇ ∇ ∇ P der , f ( t, x, x , ⋅ ; y, y )∥ L p ( µ β ) ≤ C ∥ f ∥ L p ( µ β ) ( βt ) − ε Φ C ( tβ , x − x ) Φ C ( tβ , y − y ) . Proof of Proposition 4.1.
The proof is essentially given in the paragraph preceding Proposition 4.1.We use the Feynman-Kac formula (4.7) together with the factorization formula (4.9) and the regularityestimates stated in Proposition 3.7. (cid:3) G der , f . From these esti-mates, we deduce the bounds on the elliptic Green’s matrix and its gradient stated in the following proposition.
Proposition . For any regularity exponent ε > , there exists an inverse temperature β ( d, ε ) < ∞ suchthat the following statement holds. For any inverse temperature β > β , there exists a constant C ( d, ε ) < ∞ such that for each ( x, y, x , y ) ∈ ( Z d ) , one has the estimate ∥G der , f ( x, y, ⋅ ; x , y )∥ L p ( µ β ) ≤ Cβ ∥ f ∥ L p ( µ β ) ∣ x − x ∣ d − + ∣ y − y ∣ d − . Then, for any permutation ∇ , ∇ , ∇ and ∇ of the set of gradients ∇ x , ∇ x , ∇ y and ∇ y , one has the estimates: (i) On the gradient of the Green’s matrix ∥∇ G der , f ( x, y, ⋅ ; x , y )∥ L p ( µ β ) ≤ Cβ ∥ f ∥ L p ( µ β ) ∣ x − x ∣ d − − ε + ∣ y − y ∣ d − − ε ;(ii) On the double gradient of the Green’s matrix ∥∇ ∇ G der , f ( x, y, x , y , ⋅)∥ L p ( µ β ) ≤ Cβ ∥ f ∥ L p ( µ β ) ∣ x − x ∣ d − ε + ∣ y − y ∣ d − ε ;(iii) On the triple gradient of the Green’s matrix ∥∇ ∇ ∇ G der , f ( x, y, ⋅ ; x , y )∥ L p ( µ β ) ≤ Cβ ∥ f ∥ L p ( µ β ) ∣ x − x ∣ d + − ε + ∣ y − y ∣ d + − ε ;(iv) On the quadruple gradient of the Green’s matrix ∥∇ ∇ ∇ ∇ G der , f ( x, y, ⋅ ; x , y )∥ L p ( µ β ) ≤ Cβ ∥ f ∥ L p ( µ β ) ∣ x − x ∣ d + − ε + ∣ y − y ∣ d + − ε . Proof of Proposition 4.2.
The estimates on the elliptic Green’s matrix are obtained by integratingthe inequalities of Proposition 4.1 over the times t in [ , ∞) . (cid:3) HAPTER 6
Quantitative convergence of the subadditive quantities
In this chapter, we introduce two subadditive energy quantities related to the variational formulationassociated to the Helffer-Sj¨ostrand operator described in Chapter 3. The first one, denoted by ν (◻ , p ) ,represents the energy of the minimizer associated to the Dirichlet problem in a cube ◻ with affine boundarycondition l p ( x ) ∶= p ⋅ x . The second one, denoted by ν ∗ (◻ , q ) , represents the energy of the minimizer associatedto the Neumann problem with boundary flux ∇ l q . These two quantities satisfy a subadditivity property withrespect to the domain of integration and converges as the side length of the cube tends to infinity. Moreover,the quantities ν and ν ∗ are convex with respect to the slopes of the boundary condition p and q and are insome sense convex dual to each other. The main focus of this section is then to prove by a multiscale argumentthat as the size of the domains tends to infinity, these quantities converge to a pair of dual convex conjugatefunctions and to extract from the proof a quantification of the rate of convergence.While the general strategy comes from the theory of quantitative stochastic homogenization presentedin [ ], the adaptation of the techniques presented in this monograph requires to overcome three types ofdifficulties: ● One needs to take into account the Laplacian with respect to the φ -variable; ● One needs to take into account the infinite range of the operator L ; ● We need to homogenize an elliptic system instead of an elliptic PDE.While the first point has been successfully treated in [ ] to study the ∇ φ model, the last two points are intrinsicto the Coulomb gas representation of the Villain model and will be treated in this chapter.In Section 4, we introduce a finite-volume version of first-order corrector associated to the Hellfer-Sj¨ostrandoperator L . We use the quantitative rate of convergence of the energy ν to establish quantitative sublinearityof the corrector and to prove a quantitative estimate on the weak norm of its flux. This function and itsproperties are crucial to prove the quantitative homogenization of the mixed derivative of the Green’s matrixin Chapter 7.Throughout this entire chapter, we fix a regularity exponent ε which is small compared to 1 and dependsonly on the dimension d . We assume that the inverse temperature β is large enough so that all the resultspresented in Chapter 5 hold with the regularity exponent ε .We complete this section by mentioning that in this chapter, the constants are only allowed to depend inthe dimension d as we need to be track their dependence in the inverse temperature β . The objective is toprove that the quantitative rate of convergence α obtained in Proposition 1.10 and 4.3 remains bounded awayfrom 0 as β tends to infinity.
1. Definition of the subadditive quantities and basic properties1.1. Definition of the energy quantities.
Let ◻ ⊆ Z d be a cube of Z d , we define the energy functional E ◻ according to the formula, for each function u ∈ H ( Z d , µ β ) , E ◻ [ u ] ∶= β ∑ y ∈ Z d ∥ ∂ y u ∥ L (◻ ,µ β ) + ∥∇ u ∥ L (◻ ,µ β ) + ∑ n ≥ β n ∥∇ n + u ∥ L ( Z d ,µ β ) − β ∑ supp q ∩◻≠∅ ⟨∇ q u ⋅ a q ∇ q u ⟩ µ β . We introduce the bilinear form associated to the energy E ◻ according to the formula, for each function u ∈ H ( Z d , µ β ) , B ◻ [ u, v ] ∶= β ∑ x ∈◻ ∑ y ∈ Z d ⟨ ∂ y u ( x, ⋅) , ∂ y v ( x, ⋅)⟩ µ β + ∑ x ∈◻ ⟨∇ u ( x, ⋅) , ∇ v ( x, ⋅)⟩ µ β + ∑ n ≥ ∑ x ∈ Z d β n ⟨∇ n + u ( x, ⋅) , ∇ n + v ( x, ⋅)⟩ µ β − ∑ supp q ∩◻≠∅ β ⟨∇ q u ⋅ a q ∇ q v ⟩ µ β .
878 6. QUANTITATIVE CONVERGENCE OF THE SUBADDITIVE QUANTITIES
One cannot consider the energy E ◻ of a function v only defined in the cube ◻ since the infinite range of theoperator L requires to know the value of the function on the entire space Z d . To fix this issue, we need toremove a boundary layer from a given cube ◻ . This is done in the definition below. Definition . Given a cube ◻ ∶= z + (− R , R ) d , we defined the trimmed cube ◻ − bythe formula ◻ − ∶= z + (− R + √ R , R − √ R ) d . We define the energy E ∗◻ according to the formula E ∗◻ [ u ] = β ∑ y ∈ Z d ∥ ∂ y u ∥ L (◻ ,µ β ) + ∑ n ≥ ∑ x ∈◻ , dist ( x,∂ ◻)≥ n β n ∥∇ n + u ( x, ⋅)∥ L ( µ β ) − β ∥∇ u ∥ L (◻∖◻ − ,µ β ) − β ∑ supp q ⊆◻ ⟨∇ q u ⋅ a q ∇ q u ⟩ µ β , as well as the corresponding bilinear form B ∗◻ , for each u, v ∈ H (◻ , µ β ) , B ∗◻ [ u, v ] ∶= β ∑ x ∈◻ ∑ y ∈ Z d ⟨ ∂ y u ( x, ⋅) , ∂ y v ( x, ⋅)⟩ µ β + ∑ n ≥ ∑ x ∈◻ , dist ( x,∂ ◻)≥ n β n ⟨∇ n + u ( x, ⋅) , ∇ n + v ( x, ⋅)⟩ µ β − β ∑ x ∈◻∖◻ − ⟨∇ u ( x, ⋅) , ∇ v ( x, ⋅)⟩ µ β − β ∑ supp q ⊆◻ ⟨∇ q u ⋅ a q ∇ q v ⟩ µ β . Let us make a few remarks about the definition of the energy E ∗◻ . Remark . The iterated Laplacian ∆ n has range 2 n ; given a point x ∈ ◻ , we only consider the iterationof the Laplacian until the integer n ∶= dist ( x, ∂ ◻) . This ensures that for any function v ∈ H (◻ , µ β ) , thequantity ∆ n v is well-defined. Remark . We only consider the charges q whose support is included in the cube ◻ , this ensures thatfor any function v ∈ H (◻ , µ β ) , the quantity ∇ q ⋅ a q ∇ q v is well-defined. Remark . We subtract an additional term in the boundary layer { x ∈ ◻ ∶ dist ( x, ∂ ◻) ≤ √ R } . Thisterm is a perturbative terms for two reasons: we are only summing on a small boundary layer of size √ R ofthe cube ◻ and the multiplicative factor β − is much smaller than the leading order term of the energy E ∗◻ ,which is of order 1. The reason justifying the presence of this term is that it is useful to deal with the infiniterange of the operator L ; in particular, it is useful to prove the subadditivity of the energy functional ν ∗ (seeDefinition 1.5) in Proposition 2.5. The specific choice for the exponent for the power of β is arbitrary; weonly need an exponent which is strictly between 0 and .By choosing the inverse temperature β sufficiently large, one can prove that the energy E ◻ satisfies thefollowing coercivity and boundedness properties: there exist constants c ( d ) > C ( d ) < ∞ such that, foreach u ∈ H ( Z d , µ β ) ,(1.1) c (cid:74) u (cid:75) H (◻ ,µ β ) ≤ E ◻ [ u ] ≤ C (cid:74) u (cid:75) H (◻ ,µ β ) , where we recall the notation (cid:74) u (cid:75) H (◻ ,µ β ) introduced in Section 1.5 of Chapter 2, (cid:74) u (cid:75) H (◻ ,µ β ) ∶= ⎛⎝ β ∑ y ∈ Z d ∥ ∂ y u ∥ L (◻ ,µ β ) ⎞⎠ + ∥∇ u ∥ L (◻ ,µ β ) . The same estimate holds for the energy functional E ∗◻ : for each u ∈ H (◻ , µ β ) ,(1.2) c (cid:74) u (cid:75) H (◻ ,µ β ) ≤ E ∗◻ [ u ] ≤ C (cid:74) u (cid:75) H (◻ ,µ β ) . We now proceed by giving the definitions of the subadditive quantities ν and ν ∗ . . DEFINITION OF THE SUBADDITIVE QUANTITIES AND BASIC PROPERTIES 89 Definition . Given a vector p ∈ R d ×( d ) , we write p = ( p , . . . , p ( d ) ) where the components p , . . . , p ( d ) belong to the space R d . We denote by l p the affine functiondefined by the formula l p ∶= ⎧⎪⎪⎪⎨⎪⎪⎪⎩ Z d → R ( d ) ,x → ( p ⋅ x, . . . , p ( d ) ⋅ x ) . For each cube ◻ ⊆ Z d and each pair of vectors p, p ∗ ∈ R d ×( d ) , we define the energies(1.3) ν (◻ , p ) ∶= inf u ∈ l p + H (◻ ,µ β ) ∣ ◻ ∣ E ◻ [ u ] , and(1.4) ν ∗ (◻ , p ∗ ) ∶= sup v ∈ H (◻ ,µ β ) − ∣ ◻ ∣ E ∗◻ [ v ] + ∣ ◻ ∣ ∑ x ∈◻ p ∗ ⋅ ⟨∇ v ( x )⟩ µ β . It is clear from the estimate (1.1) that the energy quantities ν and ν * are well-defined, quadratic in thevariables p and p ∗ respectively and that they satisfy the upper and lower bounds, for each cube ◻ ⊆ Z d andeach pair of vectors p, p ∗ ∈ R d ×( d ) ,(1.5) c ∣ p ∣ ≤ ν (◻ , p ) ≤ C ∣ p ∣ and c ∣ p ∗ ∣ ≤ ν (◻ , p ∗ ) ≤ C ∣ p ∗ ∣ . It follows from the standard argument of the calculus of variations that the minimizer in the variationaldefinition (1.3) exists and is unique. We denote it by u (⋅ , ◻ , p ) .The maximizer of the variational formulation (1.4) exists and is unique up to additive constant. Thisproperty is not a direct consequence of the standard arguments. It requires to use the properties of theHelffer-Sj¨ostrand equation and the regularity estimates established in Chapter 5. We postpone the proof of thisresult to Appendix B. We denote by v (⋅ , ◻ , p ∗ ) the unique maximizer which satisfies ∑ x ∈◻ ⟨ v ( x, ⋅ , ◻ , p ∗ )⟩ µ β = x ∈ ◻ and each p ∗ ∈ R d ×( d ) ,(1.6) var [ v ( x, ⋅ , ◻ n , p ∗ )] ≤ C ∣ p ∗ ∣ . The maps p ↦ u (⋅ , ⋅ , ◻ , p ) and p ∗ ↦ v (⋅ , ⋅ , ◻ , p ∗ ) are linear and that by the upper bound on the energies E and E ∗ stated in (1.1), they satisfy the estimates(1.7) ∥∇ u (⋅ , ⋅ , ◻ , p )∥ L (◻ ,µ β ) ≤ C ∣ p ∣ and ∥∇ v (⋅ , ⋅ , ◻ , p ∗ )∥ L (◻ ,µ β ) ≤ C ∣ p ∗ ∣ . The goal of this section is to prove that, as the size of the cube ◻ tends to infinity, the two quantities ν and ν ∗ converge and to obtain an algebraic rate of convergence. We obtain a result along a specific sequenceof cubes defined below. Definition Z n ) . We define the sequence l n of non-negative real numbers accordingto the induction formula l = n ∈ N , l n + = l n + √ l n . For each n ∈ N , we define the cube ◻ n ∶= (− l n , l n ) d . We denote by Z m,n ∶= l n m − n Z d ∩ ◻ n and by BL m,n the mesoscopic boundary layer defined by the formula BL m,n ∶= ◻ n ∖ ⋃ z ∈Z m,n ( z + ◻ m ) . The cube ◻ n can bepartitioned according to the formula ◻ n ∶= ⋃ z ∈Z m,n ( z + ◻ m ) ∪ BL m,n . We also introduce the notations Z n ∶= Z n,m , BL n ∶= BL n + ,n and let A n be the set(1.8) A n ∶= ⋃ z ∈Z n ( z + ◻ n ) ∖ ( z + ◻ − n ) . We refer to Figure 1 for an illustration of these definitions. The reason we introduce the sets BL m,n and A n isto treat the infinite range of the operator L .In the following remarks, we record without proof some properties pertaining the Definition 1.6. Remark . There exists a universal constant C such that, for each integer n ∈ N , 3 n ≤ l n ≤ C n . Remark . The cardinality of Z m,n is equal to 3 d ( n − m ) . Remark . One has the volume estimate ∣ BL m,n ∣ ≤ C − m ∣ ◻ n ∣ . Figure 1.
The picture on the left represents the cube ◻ n + , the white interior cubes are the cubes ( z +◻ n ) z ∈Z n and the set in black is the boundary layer BL n . The picture on the right represents the same cube; the set A n is drawn in black. The main result obtained in this chapter is a quantitative rate ofconvergence for the two energy quantities ν and ν ∗ ; it is stated below. Proposition . There exists an inverse temperature β ∶= β ( d ) < ∞ such that the following statementholds. There exist constants c ∶= c ( d ) > , C ∶= C ( d ) < ∞ and an exponent α ∶= α ( d ) > such that for eachinverse temperature β ≥ β there exists a symmetric positive definite matrix a ∈ R d ( d )× d ( d ) such that for eachinteger n ∈ N , and each pair of vectors p, p ∗ ∈ R d ×( d ) , one has the estimate ∣ ν (◻ − n , p ) − p ⋅ a p ∣ ≤ C − αn ∣ p ∣ and ∣ ν ∗ (◻ n , p ∗ ) − p ⋅ a − p ∗ ∣ ≤ C − αn ∣ p ∗ ∣ . Remark . Using the symmetries of the model, we can prove that the following properties. If we let L , d ∗ be the linear map introduced in Section 2.2 of Chapter 2, then there exists a coefficient λ β ∶= λ β ( d, β ) which tends to 0 as β tends to infinity such that(1.9) ⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩ a = I d in the space Ker L , d ∗ , a = ( + λ β ) I d in the space ( Ker L , d ∗ ) ⊥ . A direct consequence of (1.9) is the identity between the elliptic systems −∇ ⋅ a ∇ = ( d ∗ d + ( + λ β ) dd ∗ ) . These properties are a consequence of Property (3) of Proposition 1.12and Proposition 1.10.The proof of Proposition 1.10 relies on ideas which were initially developed in [ ], and follows thepresentation given in [ ]. The argument relies on the definition of the quantity(1.10) J (◻ , p, p ∗ ) ∶= ν (◻ − , p ) + ν ∗ (◻ , p ∗ ) − p ⋅ p ∗ . By the estimate (1.21) below, we know that the quadratic form J is is almost positive, in the sense that itsatisfies the inequality, for each cube ◻ of size R and each pair of slopes p, p ∗ ∈ R d ×( d ) , J (◻ , p, p ∗ ) ≥ − CR − (∣ p ∣ + ∣ p ∗ ∣ ) . To prove Proposition 1.10, we argue that the map J (◻ , p, p ∗ ) can be bounded from above in the followingsense: for each vector p ∈ R d , there exists a vector p ∗ ∈ R d such that(1.11) J (◻ , p, p ∗ ) ≤ C − αn ∣ p ∣ . Additionally, we prove that the vector p ∗ is close to a p . The quantitative rate of convergence stated inProposition 1.10 is then a relatively straightforward consequence of the estimate (1.11). The proof of (1.11) . DEFINITION OF THE SUBADDITIVE QUANTITIES AND BASIC PROPERTIES 91 relies on a hierarchical decomposition of space and requires to introduce the subadditivity defect at scale l n , τ n ∶= sup p,p ∗ ∈ B ( ν (◻ − n , p ) − ν (◻ − n + , p )) + ( ν ∗ (◻ n , p ∗ ) − ν ∗ (◻ n + , p ∗ )) (1.12) = sup p,p ∗ ∈ B J (◻ n , p, p ∗ ) − J (◻ n + , p, p ∗ ) . We then prove a series of propositions and lemmas (Propositions 2.1 and 2.5, Lemmas 3.1, 3.2, 3.5 and 3.6),where various quantities are estimated in terms if the subadditivity defect τ n . From these results we deducean inequality of the form: for each integer n ∈ N and each vector p ∈ R d ×( d ) , there exists a vector p ∗ ∈ R d ×( d ) such that J (◻ n + , p, p ∗ ) ≤ Cτ n , which can be rewritten(1.13) J (◻ n + , p, p ∗ ) ≤ CC + J (◻ n , p, p ∗ ) . The estimate (1.13) shows that, by passing from one scale to another, the energy quantity J has to contractby a multiplicative factor strictly less than 1. An iteration of the inequality (1.13) yields the algebraic rate ofconvergence stated in the inequality (1.11). We first record some basic properties of the energy quantities ν and ν ∗ ; theyare analogous to [ , Lemma 2.2]. Proposition ν and ν ∗ ) . Fix a cube ◻ ⊆ Z d and two parameters p, p ∗ ∈ R d ×( d ) .The energy quantity ν (◻ , p ) (resp. ν ∗ (◻ , p ∗ ) ) and the minimizer u (⋅ , ◻ , p ) (resp. maximizer v (⋅ , ◻ , p ∗ ) ) satisfythe properties: (1) First variation. The optimizing functions satisfy the following identities: B ◻ [ u (⋅ , ◻ , p ) , w ] = , ∀ w ∈ H (◻ , µ β ) , and B ∗◻ [ v (⋅ , ◻ , p ∗ ) , w ] = ∣ ◻ ∣ ∑ x ∈◻ p ∗ ⋅ ⟨∇ w ( x, ⋅)⟩ µ β , ∀ w ∈ H (◻ , µ β ) . (2) Second variation. For each function w ∈ l p + H (◻ , µ β ) , (1.14) 12 ∣ ◻ ∣ E ◻ [ w ] − ν (◻ , p ) = ∣ ◻ ∣ E ◻ [ u (⋅ , ◻ , p ) − w ] . For each w ∈ H (◻ , µ β ) , (1.15) ν ∗ (◻ , p ∗ ) + E ◻ [ w ] − ∣ ◻ ∣ ∑ x ∈◻ p ∗ ⋅ ⟨∇ w ( x )⟩ µ β = ∣ ◻ ∣ E ◻ [ v (⋅ , ◻ , p ∗ ) − w ] . (3) Quadratic representation. We recall the definition linear map L , d ∗ ∶ R d ×( d ) → R d introduced inSection 2.2 of Chapter 2. There exist two symmetric positive definite matrices a (◻) , a ∗ (◻) ∈ R d ( d )× d ( d ) such that (1.16) ν (◻ , p ) = p ⋅ a (◻) p and ν ∗ (◻ , p ∗ ) = p ∗ ⋅ a − ∗ (◻) p ∗ . Additionally, there exist two coefficients λ ◻ and λ ∗◻ such that (1.17) { a = I d in the space Ker L , d ∗ , a = ( + λ ◻ ) I d in the space ( Ker L , d ∗ ) ⊥ . and (1.18) { a = I d in the space Ker L , d ∗ , a = ( + λ ∗◻ ) I d in the space ( Ker L , d ∗ ) ⊥ . We denote by L t , d ∗ ∶ R d → R d ×( d ) its adjoint of the map L , d ∗ . By differentiating the identities (1.16) with respect to the parameters p and p ∗ , we obtain the equalities (1.19) 1 ∣ ◻ ∣ ∑ x ∈◻ ⎛⎝ ⟨∇ u ( x, ⋅ , ◻ , p )⟩ µ β − β ∑ supp q ∩◻≠∅ ⟨ a q ∇ q u (⋅ , ⋅ , ◻ , p )⟩ µ β L t , d ∗ ( n q ( x ))⎞⎠ = a (◻) p and (1.20) 1 ∣ ◻ ∣ ∑ x ∈◻ ⟨∇ v ( x, ⋅ , ◻ , p ∗ )⟩ µ β = a − ∗ (◻) p ∗ . (4) One-sided convex duality. For each discrete cube ◻ ⊆ Z d of sidelength R , we have the estimate (1.21) ν (◻ − , p ) + ν ∗ (◻ , p ∗ ) − p ⋅ p ∗ = ∣ ◻ ∣ E ∗◻ [ v (⋅ , ◻ , p ∗ ) − u (⋅ , ◻ − , p )] + O ( C ∣ p ∣ R − ) , where we recall the notation O introduced in Section 1 of Chapter 2: given two real numbers X, Y and a non-negative real number κ , we write X = Y + O ( κ ) if and only if ∣ X − Y ∣ ≤ κ ; Proof.
The proof of the properties (1), (2) and (3) are straightforward and we refer to [ , Lemma 2.2].For the identity (1.16), the arguments of [ ] give the following results: for each cube ◻ ⊆ Z d , there exist twopositive definite matrices a (◻) , a ∗ (◻) ∈ R d ( d )× d ( d ) , such that, for each p, p ∗ ∈ R d ×( d ) , ν (◻ , p ) = p ⋅ a (◻) p and ν ∗ (◻ , p ∗ ) = p ∗ ⋅ a ∗ (◻) p ∗ . To prove the estimate (1.17), we first use that any p ∈ Ker L , d ∗ , one has the identity d l p =
0. This implies thatthe minimizer in the energy ν (◻ , p ) is attained by the map l p , from which one obtains that the linear map a is equal to the identity on the space Ker L , d ∗ . The proof of the result on the orthogonal complement of thespace Ker L , d ∗ is a consequence of the rotation and symmetry invariance of the dual Villain model. The proofof (1.18) is identical.For the identity (1.19), by differentiating the equality (1.14) with respect to the variable p , we obtain theidentities, for each p, p ′ ∈ R d ×( d ) , a (◻) p ⋅ p ′ = ∣ ◻ ∣ ∑ x ∈◻ ⟨∇ u ( x, ⋅ , ◻ , p ) ⋅ p ′ ⟩ µ β − β ∑ supp q ∩◻≠∅ ⟨ a q ∇ q u (⋅ , ⋅ , ◻ , p )⟩ µ β ( n q , d ∗ l p ′ ) (1.22) = ∣ ◻ ∣ ∑ x ∈◻ ⟨∇ u ( x, ⋅ , ◻ , p ) ⋅ p ′ ⟩ µ β − β ∑ supp q ∩◻≠∅ ⟨ a q ∇ q u (⋅ , ⋅ , ◻ , p )⟩ µ β ( n q , L , d ∗ (∇ l p ′ ))= ∣ ◻ ∣ ∑ x ∈◻ ⟨∇ u ( x, ⋅ , ◻ , p ) ⋅ p ′ ⟩ µ β − β ∑ supp q ∩◻≠∅ ⟨ a q ∇ q u (⋅ , ⋅ , ◻ , p )⟩ µ β ( n q , L , d ∗ ( p ′ ))= ∣ ◻ ∣ ∑ x ∈◻ ⎛⎝ ⟨∇ u ( x, ⋅ , ◻ , p ) ⋅ p ′ ⟩ µ β − β ∑ supp q ∩◻≠∅ ⟨ a q ∇ q u (⋅ , ⋅ , ◻ , p )⟩ µ β L t , d ∗ ( n q ( x )) ⋅ p ′ ⎞⎠ . Using that the identity (1.22) is valid for every vector p ′ ∈ R d ×( d ) , we obtain the identity (1.19).There only remains to prove the one-sided convex duality property stated in (1.21). We apply the secondvariation formula (1.15), with the function u = u (⋅ , ◻ − , p ) and use the identity1 ∣ ◻ ∣ ∑ x ∈◻ p ∗ ⋅ ⟨∇ u ( x, ⋅ , ◻ − , p )⟩ µ β = p ⋅ p ∗ , which is a consequence of the inclusion ◻ − ⊆ ◻ and the fact that the map u belongs to the space l p + H (◻ , µ β ) .We obtain ν ∗ (◻ , p ∗ ) + ∣◻∣ E ∗◻ [ u ] − p ∗ ⋅ p = ∣ ◻ ∣ E ∗◻ [ v (⋅ , ◻ , p ∗ ) − u (⋅ , ◻ − , p ∗ )] . By definition of the function u , we have the equality ν (◻ − , p ) = ∣◻ − ∣ E ◻ − [ u ] . To prove the inequality (1.21), itis thus sufficient to prove(1.23) ∣ ∣◻ − ∣ E ◻ − [ u ] − ∣ ◻ ∣ E ∗◻ [ u ]∣ ≤ CR − . The rest of the argument is devoted to the proof of the inequality (1.23). We use the two estimates E ◻ − [ u ] ≤ C ∣ p ∣ and ∣◻ ∖ ◻ − ∣∣◻∣ ≤ C √ R to deduce(1.24) ∣ ∣◻ − ∣ E ◻ − [ u ] − ∣ ◻ ∣ E ◻ − [ u ]∣ ≤ C ∣ p ∣ √ R . . DEFINITION OF THE SUBADDITIVE QUANTITIES AND BASIC PROPERTIES 93
From the inequality (1.24), we see that to prove (1.23), it is sufficient to prove the inequality(1.25) ∣ E ∗◻ [ u ] − E ◻ − [ u ]∣ ≤ CR d − . since the volume of the cube ◻ is equal to R d . The rest of the argument is devoted to the proof of theestimate (1.25). Using the definitions of the energies E ◻ − and E ∗◻ and noting that the term involving the sumover the boundary layer { x ∈ ◻ ∶ dist ( x, ∂ ◻) ≤ R } in the definition of E ∗◻ is negative, we have the inequality ∣ E ∗◻ [ u ] − E ◻ − [ u ]∣ ≤ β ∑ y ∈ Z d ∥ ∂ y u ∥ L (◻∖◻ − ,µ β ) + ∥∇ u ∥ L (◻∖◻ − ,µ β ) + ∑ n ≥ β n ∥∇ n + u ∥ L ( Z d ∖◻ n ,µ β ) − β ∑ q ∈Q −◻ ⟨∇ q u ⋅ a q ∇ q u ⟩ µ β , where we used the notation ◻ n to denote the set ◻ n ∶= { x ∈ ◻ ∶ dist ( x, ∂ ◻) ≥ n } and the notation Q −◻ is usedto denote the set of charges Q −◻ ∶= { q ∈ Q ∶ supp q ⊆ ◻ ∖ ◻ − or ( supp q ∩ ◻ − ≠ ∅ and supp q ∩ ( Z d ∖ ◻) ≠ ∅)} . We then use the following ingredients: ● The function u is equal to the affine function l p outside the cube ◻ − . This implies the identities, foreach point x ∈ ◻ ∖ ◻ − , each point y ∈ Z d and each charge q ∈ Q such that supp q ⊆ ◻ ∖ ◻ − , ∂ y u ( x, ⋅) = , ∇ u ( x, ⋅) = p, ∇ q u = ( d ∗ l p , n q ) . From these identities, we deduce that ∑ y ∈ Z d ∥ ∂ y u ∥ L (◻∖◻ − ,µ β ) = , ∥∇ u ∥ L (◻∖◻ − ,µ β ) = ∣ p ∣ ∣ ◻ ∖ ◻ − ∣ ≤ C ∣ p ∣ R − R d . Using the estimate ∣ a q ∣ ≤ Ce − c √ β ∥ q ∥ , we obtain RRRRRRRRRRR ∑ supp q ⊆◻∖◻ − ⟨∇ q u ⋅ a q ∇ q u ⟩ µ β RRRRRRRRRRR ≤ C ∣ p ∣ ∣◻ ∖ ◻ − ∣ ≤ C ∣ p ∣ R d − . ● If a charge q ∈ Q is such that its support intersects both the cube ◻ − and the set Z d ∖ ◻ , thenits diameter must be larger than R . This implies the inequality, for any such charge q ∈ Q , ∣ a q ∣ ≤ e − c √ β ∥ q ∥ ≤ e − c √ βR . This implies ∣⟨∇ q u ⋅ a q ∇ q u ⟩ µ β ∣ ≤ e − c √ β ∥ q ∥ ∥ n q ∥ ∥∇ u ∥ L ( supp n q ,µ β ) ≤ C q e − c √ β ∥ q ∥ ∥∇ u ∥ L (◻ ,µ β ) . Summing over all the charges whose support intersects the cube ◻ − and the set Z d ∖ ◻ , we obtain ∑ q ∈Q ∣⟨∇ q u ⋅ a q ∇ q u ⟩ µ β ∣ ≤ ⎛⎝ ∑ q ∈Q e − c √ β ∥ q ∥ ∥ n q ∥ ⎞⎠ ∥∇ u ∥ L (◻) ≤ e − c √ βR ∥∇ u ∥ L (◻ ,µ β ) . We then use the upper bounds stated in (1.7) to deduce ∑ q ∈Q ∣⟨∇ q u ⋅ a q ∇ q u ⟩ µ β ∣ ≤ Ce − c √ βR R d ∥∇ u ∥ L (◻ ,µ β ) ≤ e − c √ βR ∣ p ∣ , where we reduced the value of the constant c in the second inequality to absorb the volume term R d in the exponential term e − c √ βR . ● Using that the function u is equal to the affine function outside the cube ◻ − , that for each integer n larger than 2, the iterated gradient of the affine function ∇ n l p is equal to 0 and the fact that theoperator ∇ n has range n , we obtain the identity, for each point x ∈ Z d such that dist ( x, ◻ − ) ≥ n , ∇ n u ( x ) =
0. From this identity, we deduce(1.26) ∑ n ≥ β n ∥∇ n u ∥ L ( Z d ∖◻ n ,µ β ) ≤ ∑ n ≥ √ R ∑ dist ( x, ◻ − )≤ n β n ∥∇ n u ( x, ⋅)∥ L ( µ β ) . We then estimate the L -norm of the iterated gradient ∇ n according to the estimate, for each x ∈ Z d , ∥∇ n u ( x, ⋅)∥ L ( µ β ) ≤ C n ∥∇ u ∥ L ( B ( x,n ) µ β ) ≤ C n (∥∇ u ∥ L ( B ( x,n )∩◻ ,µ β ) + ∥∇ l p ∥ L ( B ( x,n )∩( Z d ∖◻) ,µ β ) ) (1.27) ≤ C n ( R d ∥∇ u ∥ L (◻ ,µ β ) + n d ∣ p ∣ )≤ C n R d ∣ p ∣ , where we used the estimate (1.7) in the last inequality. A combination of the inequalities (1.26)and (1.27) with the upper bound (1.7) shows ∑ n ≥ β n ∥∇ n u ∥ L ( Z d ∖◻ n ,µ β ) ≤ ∑ n ≥ √ R ∣{ x ∈ Z d ∶ dist ( x, ◻ − ) ≤ n }∣ C n R d β n ∣ p ∣ . The volume factor ∣{ x ∈ Z d ∶ dist ( x, ◻ − ) ≤ n }∣ can be estimated by the value ( R + n ) d . Thus, bychoosing the inverse temperature β large enough, we obtain that there exists a constant c ∶= c ( d ) > ∑ n ≥ β n ∥∇ n u ∥ L ( Z d ∖◻ n µ β ) ≤ ( C √ β ) √ R R d ∣ p ∣ ≤ Ce − c ( ln β )√ R ∣ p ∣ . A combination of the three previous items completes the proof of completes the proof the inequality (1.21). (cid:3)
2. Subadditivity for the energy quantities
In this section, we prove a subadditivity property for the two energies ν and ν ∗ . The result is quantifiedand we estimate the H -norm of the difference of the minimizer u (resp. maximizer v ) over two different scalesin terms of the difference ν (◻ m , p ) − ν (◻ n , p ) (resp. ν ∗ (◻ m , p ∗ ) − ν ∗ (◻ n , p ∗ ) ). ν . In this section, we prove that the energy quantity ν satisfiesa subadditivity property with respect to the domain of integration and deduce from it the existence of thehomogenized coefficient a . The statement of Proposition 2.1 is quantified; we prove that the H -norm of thedifference of the minimizer u over two different scales in terms of the subadditivity defect for the energy ν . Proposition ν ) . There exists an inverse temperature β ∶= β ( d ) < ∞ such thatfor each β ≥ β the following statement is valid. There exists a constant C ∶= C ( d ) < ∞ such that for each pairof integers ( m, n ) ∈ N satisfying n > m and each vector p ∈ R d ×( d ) , (2.1) 1 ∣Z m,n ∣ ∑ z ∈Z m,n (cid:74) u (⋅ , ◻ n , p ) − u (⋅ , z + ◻ m , p ) (cid:75) H (◻ n + ,µ β ) ≤ C ( ν (◻ m , p ) − ν (◻ n , p ) + C − m ∣ p ∣ ) . Remark . Since it is useful in the rest of the proof, we note that the demonstration of Proposition 2.1can be adapted to the case of trimmed cubes so as to obtain the estimate, for each pair of integers m, n ∈ N such that m ≤ n ,1 ∣Z m,n ∣ ∑ z ∈Z m,n (cid:74) u (⋅ , ◻ − n , p ) − u (⋅ , z + ◻ − m , p ) (cid:75) H (◻ n + ,µ L ) ≤ C ( ν (◻ − m , p ) − ν (◻ − n , p ) + C − m ∣ p ∣ ) . Since the proof is essentially the same as the proof of Proposition 2.1; the details are left to the reader.Before proving Proposition 2.1, we record an immediate corollary of the the subadditivity property for theenergy ν . Corollary . There exists an inverse temperature β ∶= β ( d ) < ∞ such that, for each β ≥ β , thereexists a non-negative real number a such that for each vector p ∈ R d ×( d ) , one has ν (◻ n , p ) —→ n →∞ p ⋅ a p. By Property (3) of Proposition 1.12, this statement can be rewritten equivalently as a (◻ n ) —→ n →∞ a . Additionally, one deduces from (2.1) the lower bound estimate in the sense of symmetric positive definitematrices (2.2) a (◻ n ) ≥ a − C − n . . SUBADDITIVITY FOR THE ENERGY QUANTITIES 95 Remark . By Remark 2.2, the convergence also holds with the trimmed triadic cubes and we have, foreach vector p ∈ R d ×( d ) , ν (◻ − n , p ) —→ n →∞ p ⋅ a p, a (◻ − n ) —→ n →∞ a and ∀ n ∈ N , a (◻ − n ) ≥ a − C − n . Proof.
Since the left side of (2.1) is non-negative, we have the inequality, for each pair of integers m, n ∈ N such that n > m (2.3) ν (◻ n , p ) ≤ ν (◻ m , p ) + C − m ∣ p ∣ . Combining the inequality (2.3) with the fact that the sequence ( ν (◻ n , p )) n ∈ N is non-negative implies that itconverges with the estimate (2.2). (cid:3) We now focus on the proof of Proposition 2.1.
Proof of Proposition 2.1.
For the sake of simplicity, we only write the proof in the case when thedifference between the integers m and n is equal to 1: we consider the specific case of the pair ( n, n + ) . Theproof of the general case is similar. We assume without loss of generality that ∣ p ∣ = . We let w be the function of l p + H (◻ n + , µ β ) defined by the following construction: ● For each point z ∈ Z n + , we set w ∶= u (⋅ , z + ◻ n , p ) ; ● On the mesoscopic boundary layer BL n , we set w ∶= l p .Applying the second variation formula (1.14) and the coercivity of the energy functional E stated in (1.1)gives the inequality(2.4) (cid:74) u (⋅ , ◻ n + , p ) − w (cid:75) H (◻ n + ,µ β ) ≤ C ( ∣ ◻ n + ∣ E ◻ n + [ w ] − ν (◻ , p )) . Using that, for each point z ∈ Z n , the function w is equal to the minimizer u (⋅ , z + ◻ n , p ) in the cube ( z + ◻ n ) ,we have the inequality(2.5) ∑ z ∈Z n + (cid:74) u (⋅ , ◻ n + , p ) − u (⋅ , z + ◻ n , p ) (cid:75) H (◻ n + ,µ β ) ≤ (cid:74) u (⋅ , ◻ n + , p ) − w (cid:75) H (◻ n + ,µ β ) . By the estimates (2.4) and (2.5), we see that to prove the inequality (2.1), it is thus sufficient to prove(2.6) 12 ∣ ◻ ∣ E ◻ n + [ w ] ≤ ν (◻ n , p ) + C − n . We now prove the inequality (2.6). By definition of the energy E , we have(2.7) E ◻ n + [ w ] ∶= β ∑ y ∈ Z d ∥ ∂ y w ∥ L (◻ n + ,µ β ) ·„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„¶ (2.7) −( i ) + ∥∇ w ∥ L (◻ n + ,µ β ) ·„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„¶ (2.7) −( ii ) + ∑ k ≥ β k ∥∇ k + w ∥ L ( Z d ,µ β ) ·„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„¶ (2.7) −( iii ) − β ∑ supp q ∩◻ n + ≠∅ ⟨∇ q w ⋅ a q ∇ q u ⟩ µ β ·„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„¶ (2.7) −( iv ) . We estimate the four terms on the right side separately. The term (2.7)-(i) involving the derivative with respectto the field φ can be estimated by the following argument. Since the map w is equal to the deterministic affinefunction l p in the boundary layer BL n , we have the identity ∂ y w ( x, ⋅) = x ∈ BL n and any point y ∈ Z d . This implies the equality(2.8) ∑ y ∈ Z d ∥ ∂ y w ∥ L (◻ n + µβ ) = ∑ z ∈Z n ∑ y ∈ Z d ∥ ∂ y u (⋅ , z + ◻ n )∥ L ( z +◻ n µ β ) . This completes the estimate of the term (2.7)-(i). For the term (2.7)-(ii), we use the same argument and notethat ∇ w ( x, ⋅) = p for any point x ∈ BL n . We obtain1 ∣◻ n + ∣ ∥∇ w ∥ L (◻ n + µ β ) = ∣ ◻ n + ∣ ∑ z ∈Z n ∥∇ u (⋅ , z + ◻ n , p )∥ L ( z +◻ n ,µ β ) + ∣ BL n ∣∣◻ n + ∣ (2.9) ≤ ∣ ◻ n + ∣ ∑ z ∈Z n ∥∇ u (⋅ , z + ◻ n , p )∥ L ( z +◻ n ,µ β ) + C − n . To estimate the third term (2.7)-(iii), we note that, by Remark 1.7, the boundary layer BL n has a width oforder c n , where c is a universal constant. We split the sum over the integer k at the value c n and write(2.10) ∑ k ≥ β k ∥∇ k + w ∥ L ( Z d µ β ) = ∑ ≤ k ≤ c n β k ∥∇ k + w ∥ L ( Z d ,µ β ) ·„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„¶ (2.10) −( i ) + ∑ k > c n β k ∥∇ k + w ∥ L ( Z d ,µ β ) ·„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„¶ (2.10) −( ii ) , and estimate the two terms in the right side separately. For the term (2.10)-(i), we use the following ingredients: ● The boundary layer BL n has width c n ; ● For each integer k ∈ N , the iterated gradient ∇ k has range k ; ● For each integer k ≥
2, the k th-iterated gradient of the affine function l p is equal to 0, i.e., ∇ k l p = ● The function w is equal to the affine function l p in the boundary layer BL n and, for each point z ∈ Z n , is equal to the minimizer u (⋅ , z + ◻ n , p ) in the subcube ( z + ◻ n ) .We obtain the identity(2.11) ∑ ≤ k ≤ c n β k ∥∇ k + w ∥ L ( Z d ,µ β ) = ∑ z ∈Z n ∑ ≤ k ≤ c n β k ∥∇ k + u (⋅ , z + ◻ n , p )∥ L ( Z d ,µ β ) . The term (2.10)-(ii) is an error term which is small. Using that the the discrete gradient is a bounded operatorwhich has range k and that the function w is equal to the affine function l p outside the cube ◻ n + , we write ∑ k > c n β k ∥∇ k + w ∥ L ( Z d ,µ β ) ≤ ∑ k > c − n C k β k ∥∇ w ∥ L (◻ n + ,µ β ) ≤ ∑ k > c n C k ∣◻ n + ∣ β k ∥∇ w ∥ L (◻ n + ,µ β ) . We then use that the volume of the cube ◻ n + is smaller than the value C dn and the upper bound ∥∇ w ∥ L (◻ n + ,µ β ) ≤ C which is a consequence of the upper bound (1.7) and the the definition of the map w . This argument yields, if the inverse temperature β is chosen large enough,(2.12) ∑ k > c n β k ∥∇ k + w ∥ L ( Z d ,µ β ) ≤ ⎛⎝ ∑ k > c n C k β k ⎞⎠ dn ≤ ( C √ β ) c n dn ≤ Ce − c ( ln β ) − n . With the same argument, we obtain the estimate, for each point z ∈ Z n ,(2.13) ∑ k > c n β k ∥∇ k + u (⋅ , z + ◻ n , p )∥ L ( Z d ,µ β ) ≤ Ce − c ( ln β ) − n . By combining the estimates (2.11), (2.12) and (2.13), we have obtained the upper bound(2.14) ∑ k ≥ β k ∥∇ k + w ∥ L ( Z d ,µ β ) ≤ ∑ z ∈Z n ∑ k ≥ β k ∥∇ k + u (⋅ , z + ◻ n , p )∥ L ( Z d ,µ β ) + Ce − c ( ln β ) n . This completes the estimate of the term term (2.7)-(iii).The term (2.7)-(iv) can be estimated similarly, we partition the set of charges q whose support intersectsthe cube ◻ n + into three subsets: ● The set of charges whose support does not intersect any cube of the collection ( z + ◻ n ) z ∈Z n . Wedenote this set by Q ; ● The set of charges q whose support intersects exactly one cube of the collection ( z + ◻ n ) z ∈Z n . Wedenote this set by Q and for z ∈ Z n , we denote by Q ,z the set of charges whose support onlyintersects the cube ( z + ◻ n ) ; ● The set of charges q whose support intersects at least two cubes of the collection ( z + ◻ n ) z ∈Z n .We denote this set by Q and note that a charge belonging to this set must satisfy the propertydiam q ≥ c − n .We then partition the sum(2.15) ∑ supp q ∩◻ n + ≠∅ ⟨∇ q w ⋅ a q ∇ q w ⟩ µ β = ∑ q ∈Q ⟨∇ q w ⋅ a q ∇ q w ⟩ µ β ·„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„¶ (2.15) −( i ) + ∑ q ∈Q ⟨∇ q w ⋅ a q ∇ q u ⟩ µ β ·„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„¶ (2.15) −( ii ) + ∑ q ∈Q ⟨∇ q w ⋅ a q ∇ q w ⟩ µ β ·„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„¶ (2.15) −( iii ) . . SUBADDITIVITY FOR THE ENERGY QUANTITIES 97 We then estimate the three terms in the right side separately. For the term (2.15)-(i), we have, by thedefinitions of the function w and of the set Q , ∑ q ∈Q ⟨∇ q w ⋅ a q ∇ q w ⟩ µ β = ∑ q ∈Q ⟨∇ q l p ⋅ a q ∇ q l p ⟩ µ β ≤ ∑ q ∈Q e − c √ β ∥ q ∥ ∥ n q ∥ L . Using that the volume of the boundary layer satisfies ∣ BL n ∣ ≤ C − n ∣◻ n + ∣ and the estimate, for each x ∈ Z d , ∑ q ∈Q e − c √ β ∥ q ∥ ∥ n q ∥ L { x ∈ supp n q } ≤ C, we obtain that(2.16) ∑ q ∈Q e − c √ β ∥ q ∥ ∥ n q ∥ L ≤ C − n ∣◻ n + ∣ . We then estimate the term (2.15)-(ii). By definition of the function w , we can write(2.17) ∑ q ∈Q ⟨∇ q w ⋅ a q ∇ q w ⟩ µ β = ∑ z ∈Z n ∑ q ∈Q ,z ⟨∇ q u (⋅ , ⋅ , z + ◻ n , p ) ⋅ a q ∇ q u (⋅ , ⋅ , z + ◻ n , p )⟩ µ β . To estimate the term (2.15)-(iii), we use the upper bound ∥∇ w ∥ L (◻ n + ,µ β ) ≤ C ∣ p ∣ on the average L -norm ofthe gradient of w , the fact that the function w is equal to the affine function l p outside the cube ◻ n + , theestimate (1.9) on the coefficients a q and the property, for each charge q ∈ Q , diam q ≥ c n . We obtain(2.18) RRRRRRRRRRR ∑ supp q ∈Q ⟨∇ q w ⋅ a q ∇ q w ⟩ µ β RRRRRRRRRRR ≤ e − c √ β n . With the same argument, we obtain the following estimate: for each z ∈ Z n ,(2.19) RRRRRRRRRRRR ∑ supp q ∩ z +◻ n ≠∅ ,q ∉Q ,z ⟨∇ q u (⋅ , ⋅ , z + ◻ n , p ) ⋅ a q ∇ q u (⋅ , ⋅ , z + ◻ n , p )⟩ µ β RRRRRRRRRRRR ≤ e − c √ β n . Combining the identity (2.15) with the inequalities (2.16), (2.17) (2.18) and (2.19) shows the estimate(2.20) ∑ supp q ∩◻ n + ≠∅ ⟨∇ q w ⋅ a q ∇ q w ⟩ µ β ≤ ∑ z ∈Z n ∑ supp q ∩( z +◻ n ) ⟨∇ q u (⋅ , z + ◻ n , p ) ⋅ a q ∇ q u (⋅ , z + ◻ n , p )⟩ µ β + C − n ∣ ◻ n + ∣ . We finally combine the equality (2.7), the estimates (2.8), (2.9), (2.14) (2.20) to obtain the inequality (2.6).The proof of Proposition 2.1 is complete. (cid:3) ν ∗ . In this section, we prove a similar statement for the energy ν ∗ . Proposition ν ∗ ) . There exists a constant C ∶= C ( d ) < ∞ such that for each pairof integers ( n, m ) ∈ N such that n > m and each vector p ∗ ∈ R d ×( d ) , (2.21) 1 ∣Z m,n ∣ ∑ z ∈Z m,n (cid:74) v (⋅ , ⋅ , ◻ n , p ∗ ) − v (⋅ , ⋅ , z + ◻ m , p ∗ ) (cid:75) H ( z +◻ m ,µ β ) ≤ C ( ν ∗ (◻ m , p ) − ν ∗ (◻ n , p ) + − n ∣ p ∗ ∣ ) . As it was the case for the energy quantity ν , we deduce from Proposition 2.5 that the sequence ( ν ∗ (◻ n , p ∗ )) n ∈ N converges as n tends to infinity. Corollary . There exists an inverse temperature β ∶= β ( d ) < ∞ such that for each β ≥ β thefollowing statement is valid. There exists a non-negative real number a ∗ such that for each vector p ∗ ∈ R d ×( d ) ,one has ν ∗ (◻ n , p ∗ ) —→ n →∞ a − ∗ ∣ p ∗ ∣ . By the Property (3) of Proposition 1.12, this statement can be rewritten equivalently a ∗ (◻ n ) − —→ n →∞ a − ∗ . We also have the lower bound, for each integer n ∈ N , a ∗ (◻ n ) − ≥ a − ∗ − C − n . Proof of Proposition 2.5.
For the sake of simplicity, we only write the proof in the case when thedifference between the integers m and n is equal to 1. We consider the specific case of the pair ( n + , n ) . Theproof of the general case is similar. We assume without loss of generality that ∣ p ∗ ∣ = v ∶= v (⋅ , ◻ n + , p ∗ ) and, for z ∈ Z n , we restrict it to the cubes ( z + ◻ n ) . We applythe second variation formula (1.15) and the coercivity of the energy functional E ∗ z +◻ n . We obtain, for eachpoint z ∈ Z n , (cid:74) v (⋅ , ◻ n + , p ∗ ) − v (⋅ , z + ◻ n , p ∗ ) (cid:75) H ( z +◻ n ,µ β ) ≤ C ( ν ∗ ( z + ◻ n , p ∗ ) + ∣ ◻ n ∣ E ∗ z +◻ n [ v ] + ∣ ◻ n ∣ ∑ x ∈ z +◻ p ∗ ⋅ ⟨∇ v ( x )⟩ µ β ) . Summing over the points z ∈ Z n and dividing by the cardinality of Z n shows1 ∣Z n ∣ ∑ z ∈Z n (cid:74) v (⋅ , ◻ n + , p ∗ ) − v (⋅ , z + ◻ n , p ∗ ) (cid:75) H ( z +◻ n ,µ β ) ≤ C ⎛⎝ ν ∗ (◻ n , p ∗ ) + ∑ z ∈Z n ∣Z n ∣ ⋅ ∣◻ n ∣ E ∗ z +◻ n [ v ] + ∣Z n ∣ ⋅ ∣◻ n ∣ ∑ x ∈◻ n p ∗ ⋅ ⟨∇ v ( x )⟩ µ β ⎞⎠ . The factor ∣Z n ∣ = d on the left side depends only on the dimension d and can thus be absorbed in the constant C in the right side. We deduce that, to prove the inequality (2.21), it is sufficient to prove(2.22) ∑ z ∈Z n ∣Z n ∣ ⋅ ∣◻ n ∣ ( E ∗ z +◻ n [ v ] + ∑ x ∈ z +◻ n p ∗ ⋅ ⟨∇ v ( x )⟩ µ β )≤ ∣ ◻ n + ∣ E ∗◻ n + [ v ]·„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„¶ (2.22) −( i ) + ∣ ◻ n + ∣ ∑ x ∈◻ n + p ∗ ⋅ ⟨∇ v ( x )⟩ µ β ·„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„¶ (2.22) −( ii ) + C − n . We first estimate the term (2.22)-(ii). We use the estimate (1.7) on the L -norm of the gradient of thefunction v , the Cauchy-Schwarz inequality and the volume estimate(2.23) ∣◻ n + ∣ − ∣Z n ∣ ⋅ ∣◻ n ∣ = ∣◻ n + ∖ ⋃ z ∈Z n ( z + ◻ n )∣ = ∣ BL n ∣ ≤ C − n ∣◻ n + ∣ . We obtain ∑ z ∈Z n ∣Z n ∣ ⋅ ∣◻ n ∣ ∑ x ∈ z +◻ n p ∗ ⋅ ⟨∇ v ( x, ⋅)⟩ µ β ≤ ∣◻ n + ∣ ∑ z ∈Z n ∑ x ∈ z +◻ n p ∗ ⋅ ⟨∇ v ( x )⟩ µ β + ( ∣ BL n ∣∣◻ n + ∣ ) ∥∇ v ∥ L (◻ n + ,µ β ) (2.24) ≤ ∣◻ n + ∣ ∑ x ∈◻ n + p ∗ ⋅ ⟨∇ v ( x, ⋅)⟩ µ β + ∣◻ n + ∣ ∑ x ∈ BL n p ∗ ⋅ ⟨∇ v ( x, ⋅)⟩ µ β + C − n ≤ ∣◻ n + ∣ ∑ x ∈◻ n + p ∗ ⋅ ⟨∇ v ( x, ⋅)⟩ µ β + ( ∣ BL n ∣∣ ◻ n + ∣ ) ∥∇ v ∥ L (◻ n + ,µ β ) + C − n ≤ ∣◻ n + ∣ ∑ x ∈◻ n + p ∗ ⋅ ⟨∇ v ( x, ⋅)⟩ µ β + C − n . To estimate the term pertaining to the energy functional E ∗ , we introduce two notations: ● We let Q n + the set of charges whose support is included in the cube ◻ n + and intersects the boundarylayer BL n ; ● For each integer k ∈ N , we let C k be the set C k ∶= { x ∈ ◻ n + ∶ B ( x, k ) ⊆ ◻ n + and B ( x, k ) /⊆ ⋃ z ∈Z n ( z + ◻ n )} . . SUBADDITIVITY FOR THE ENERGY QUANTITIES 99 We note that we have the inclusion of the boundary layers ◻ n + ∖ ◻ − n + ⊆ BL n . Thus, by the definition of theenergy functional E ∗◻ n + , we have the inequality E ∗◻ n + [ v ] ≥ β ∑ y ∈ Z d ∥ ∂ y v ∥ L (◻ ,µ β ) + ( − β ) ∥∇ v ∥ L ( BL n ,µ β ) + ∑ z ∈Z n ∥∇ v ∥ L ( z +◻ n ,µ β ) + ∑ n ≥ ∑ dist ( x,∂ ◻)≥ n β n ∥∇ n + v ( x, ⋅)∥ L ( µ β ) − β ∑ supp q ⊆◻ ⟨∇ q v ⋅ a q ∇ q v ⟩ µ β . We choose the inverse temperature β enough so that, we have the estimate − β − ≥ . We recall the definitionof the set A n stated in (1.8) and use the definition of the energy E ∗ z +◻ n to obtain the estimate E ∗◻ n + [ v ] − ∑ z ∈Z n E ∗ z +◻ n [ v ] ≥ β ∑ y ∈ Z d ∥ ∂ y v ∥ L ( BL n ,µ β ) + ∥∇ v ∥ L ( BL n ,µ β ) + β ∥∇ v ∥ L ( A n µ β ) (2.25) − β ∑ supp q ∈Q n + ⟨∇ q v ⋅ a q ∇ q v ⟩ µ β + ∑ k ≥ β k ∥∇ k + v ∥ L ( C k ,µ β ) . ≥ ∥∇ v ∥ L ( BL n ,µ β ) + β ∥∇ v ∥ L ( A n µ β ) − β ∑ supp q ∈Q n + ⟨∇ q v ⋅ a q ∇ q v ⟩ µ β ·„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„¶ (2.25) −( i ) + ∑ k ≥ β k ∥∇ k + v ∥ L ( C k ,µ β ) ·„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„¶ (2.25) −( ii ) . We first estimate the term (2.25)-(i). To this end, we partition the set Q n + into two sets of charges Q n + , and Q n + , defined by the formulas Q n + , ∶= { q ∈ Q n + ∶ supp q ⊆ BL n ∪ A n } and Q n + , ∶= Q n + ∖ Q n + , . We then use the two following arguments. First, if a charge belongs to the set Q n + , , then its diameter has tobe larger larger than c − n . By the estimate ∣ a q ∣ ≤ Ce − c √ β ∥ q ∥ , we obtain RRRRRRRRRRRR ∑ q ∈Q n + , ⟨∇ q v ⋅ a q ∇ q v ⟩ µ β RRRRRRRRRRRR ≤ ∑ q ∈Q n + , e − c √ β ∥ q ∥ ∥ n q ∥ ∥∇ v ∥ L ( supp n q ,µ β ) ≤ e − c √ β n ∥∇ v ∥ L (◻ n + ,µ β ) . We then use the estimate (1.7) to bound the L -norm of the gradient of v . We obtain(2.26) RRRRRRRRRRRR ∑ supp q ∈Q n + , ⟨∇ q v ⋅ a q ∇ q v ⟩ µ β RRRRRRRRRRRR ≤ C ∣◻ n + ∣ e − c n ≤ Ce − c √ β n , by reducing the value of the constant c in the second inequality.Second, for the charges belonging to the set Q n + , , using the estimate ∣ a q ∣ ≤ Ce − c √ β ∥ q ∥ , we have(2.27) RRRRRRRRRRRR ∑ q ∈Q n + , ⟨∇ q v ⋅ a q ∇ q v ⟩ µ β RRRRRRRRRRRR ≤ Ce − c √ β ∥∇ v ∥ L ( BL n ∪ A n ,µ β ) . By combining (2.26) and (2.27), we obtain(2.28)
RRRRRRRRRRR ∑ q ∈Q ⟨∇ q v ⋅ a q ∇ q v ⟩ µ β RRRRRRRRRRR ≤ Ce − c √ β ∥∇ v ∥ L ( BL n ∪ A n ,µ β ) + Ce − c √ β n . The term (2.25)-(ii) can be estimated with similar arguments: we need to decompose over the integers k such that C k ⊆ BL n ∪ A n and the integers k such that C k /⊆ BL n ∪ A n . Since the argument is almost the same,we omit it and only give the result. We obtain the inequality(2.29) 12 ∑ k ≥ β k ∥∇ k v ∥ L ( C k ,µ β ) ≤ Cβ ∥∇ v ∥ L ( BL n ∪ A n ,µ β ) + Ce − c ( ln β ) n .
00 6. QUANTITATIVE CONVERGENCE OF THE SUBADDITIVE QUANTITIES
We now combine the estimates (2.28) and (2.29) and deduce that ∣ (2.25)-(i) ∣ + ∣ (2.25)-(ii) ∣ ≤ ( Cβ + Ce − c √ β ) ∥∇ v ∥ L ( BL n ∪ A n ,µ β ) + C ( e − c √ β n + e − c ( ln β ) n ) . As a consequence, if β is chosen large enough depending only on the dimension d , then we have(2.30) ∣ (2.25)-(i) ∣ + ∣ (2.25)-(ii) ∣ ≤ ∥∇ v ∥ L ( BL n ,µ β ) + β ∥∇ v ∥ L ( A n µ β ) + Ce − c ( ln β ) n . Combining the estimates (2.25) and (2.30) shows E ◻ n + [ v ] − ∑ z ∈Z n E z +◻ n [ v ] ≥ − Ce − c ( ln β ) n . Dividing the previous display by 2 ∣Z n ∣ ⋅ ∣◻ n ∣ shows the estimate(2.31) ∑ z ∈Z n ∣Z n ∣ ⋅ ∣◻ n ∣ E z +◻ n [ v ] ≤ ∣Z n ∣ ⋅ ∣◻ n ∣ E ◻ n + [ v ] + Ce − c n . We use the volume estimate (2.23), the bound on the average L -norm of the gradient of v stated in (1.7) andthe coercivity of the energy functional E ∗ stated in (1.2) to deduce(2.32) 12 ∣Z n ∣ ⋅ ∣◻ n ∣ E ◻ n + [ v ] ≤ ∣◻ n + ∣ E ◻ n + [ v ] + C − n . Combining the estimates (2.31), (2.32) and (2.24) shows the inequality (2.22) and completes the proof ofProposition 2.5. (cid:3)
3. Quantitative convergence of the subadditive quantities
In this section, we prove an algebraic rate of convergence for the quantity J defined in (1.10). We recallthe definition of the subadditivity defect τ n given in (1.12) and we introduce the following notation: for eachinteger n ∈ N , we denote by(3.1) a n ∶= a ∗ (◻ n ) , and call the matrix a n the approximate homogenized matrix . We first prove a series of lemmas, estimatingvarious quantities in terms of the subadditivity defect τ n following the strategy described in Section 1.2.Before starting the proofs, let us make the following remark: By Corollaries 2.3 and 2.6, the subadditivitydefect τ n converges to 0 as n tends to infinity. In particular all the quantities which are bounded from aboveby the subadditivity defect τ n tend to 0 when n tends to infinity. The first lemma we prove establishesthat the difference between the coefficients a n over two different scales can be estimated in terms of thesubadditivity defect τ n . Lemma . There exists a constant C ∶= C ( d ) < ∞ such that for any pair of integers ( m, n ) ∈ N with m ≤ n , the following estimate holds ∣ a − n − a − m ∣ ≤ n ∑ k = m τ k + C − m . Proof.
Before starting the proof, we collect a few ingredients and notations used in the argument: ● We recall the notation O introduced in Section 1 of Chapter 2, given two real numbers X, Y and anon-negative real number κ , we write X = Y + O ( κ ) if and only if ∣ X − Y ∣ ≤ κ ; ● By the formula (1.20), we have the identity ∑ x ∈◻ n ⟨∇ v ( x, ⋅ , ◻ n , p ∗ )⟩ µ β = a − n p ∗ ; ● We recall the definition of the set Z m,n ∶= l n m − n Z d ∩ ◻ n , the definition of the boundary layer BL m,n given in Definition 1.6 and the volume estimate ∣ BL m,n ∣ ≤ C − m ∣◻ n ∣ stated in Remark 1.9; ● By definition of the subadditivity defect τ k , we have the identity, for each p ∈ R d ( d ) , ν ∗ (◻ m , p ) − ν ∗ (◻ n , p ) ≤ ∣ p ∣ n ∑ k = m τ k . . QUANTITATIVE CONVERGENCE OF THE SUBADDITIVE QUANTITIES 101 We fix a vector p ∗ ∈ R d ( d ) such that ∣ p ∗ ∣ = a − n p ∗ = ∣ ◻ n ∣ ∑ x ∈◻ n ⟨∇ v ( x, ⋅ , ◻ n , p ∗ )⟩ µ β (3.2) = ∣ ◻ n ∣ ∑ z ∈Z m,n ∑ x ∈ z +◻ m ⟨∇ v ( x, ⋅ , ◻ n , p ∗ )⟩ µ β ·„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„¶ (3.2) −( i ) + ∣ ◻ n ∣ ∑ x ∈ BL m,n ⟨∇ v ( x, ⋅ , ◻ n , p ∗ )⟩ µ β ·„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„¶ (3.2) −( ii ) . The term (3.2)-(ii) is the simplest one, we estimate it by the Cauchy-Schwarz inequality, the estimate on the L -norm of the gradient of v stated in (1.7) and the volume estimate ∣ BL m,n ∣ ≤ C − m ∣◻ n ∣ . We obtain(3.3) RRRRRRRRRRRR ∣◻ n ∣ ∑ x ∈ BL m,n ⟨∇ v ( x, ⋅ , ◻ n , p ∗ )⟩ µ β RRRRRRRRRRRR ≤ C − m . To estimate the term (3.2)-(i), we use the estimate (1.7), the identity BL m,n = ◻ n ∖ ⋃ z ∈Z m,n and the volumeestimate ∣ BL m,n ∣ ≤ C − m ∣◻ n ∣ . We obtain1 ∣ ◻ n ∣ ∑ z ∈Z n ∑ x ∈ z +◻ m ⟨∇ v ( x, ⋅ , ◻ n , p ∗ )⟩ µ β = ∣Z m,n ∣ ∣ z + ◻ m ∣ ∑ z ∈Z n ∑ x ∈ z +◻ m ⟨∇ v ( x, ⋅ , ◻ n , p ∗ )⟩ µ β + O ( C − m ) . Applying the subadditivity estimate stated in Proposition 2.1, we find that1 ∣Z m,n ∣ ∣ z + ◻ m ∣ ∑ z ∈Z n ∑ x ∈ z +◻ m ∣⟨∇ v ( x, ⋅ , ◻ n , p ∗ ) − ∇ v ( x, ⋅ , z + ◻ m , p ∗ )⟩ µ β ∣ (3.4) ≤ ∣Z m,n ∣ ∑ z ∈Z m,n ∥∇ v (⋅ , ◻ n , p ∗ ) − ∇ v (⋅ , z + ◻ m , p ∗ )∥ L ( z +◻ m ,µ β ) ≤ ⎛⎝ ∣Z m,n ∣ ∑ z ∈Z m,n ∥∇ v (⋅ , ◻ n , p ∗ ) − ∇ v (⋅ , z + ◻ m , p ∗ )∥ L ( z +◻ m ,µ β ) ⎞⎠ ≤ ⎛⎝ ∣Z m,n ∣ ∑ z ∈Z m,n (cid:74) ∇ v (⋅ , ◻ n , p ∗ ) − ∇ v (⋅ , z + ◻ m , p ∗ ) (cid:75) H ( z +◻ m ,µ β ) ⎞⎠ ≤ C ( n ∑ k = m τ k ) + C − m . We use the inequality (3.4), the translation invariance of the measure µ β and the identity ∑ x ∈◻ n ⟨∇ v ( x, ⋅ , ◻ m , p ∗ )⟩ µ β = a − m p ∗ . We obtain1 ∣Z m,n ∣ ∣ z + ◻ m ∣ ∑ z ∈Z n ∑ x ∈ z +◻ m ⟨∇ v ( x, ⋅ , ◻ n , p ∗ )⟩ µ β (3.5) = ∣Z m,n ∣ ∑ z ∈Z m,n ∣ z + ◻ m ∣ ∑ x ∈ z +◻ m ⟨∇ v ( x, ⋅ , z + ◻ m , p ∗ )⟩ µ β + O ⎛⎝ C ( n ∑ k = m τ k ) + C − m ⎞⎠= ∣◻ m ∣ ∑ x ∈◻ m ⟨∇ v ( x, ⋅ , ◻ m , p ∗ )⟩ µ β + O ⎛⎝ C ( n ∑ k = m τ k ) + C − m ⎞⎠= a − m p ∗ + O ⎛⎝ C ( n ∑ k = m τ k ) + C − m ⎞⎠ . We then combine the identity (3.2) with the estimates (3.3) and (3.5) to complete the proof of Lemma 3.1. (cid:3) v . The next step in theargument is to control the variance of the spatial average of the maximiser v . We prove that its variancecontracts and obtain an algebraic rate of convergence. The proof relies on an explicit computation and makesuse of the differentiated Helffer-Sj¨ostrand equation introduced in Section 4 of Chapter 3 to estimate thecorrelation between the random variables φ ↦ v ( x, φ, ◻ n + , p ) and φ ↦ v ( x ′ , φ, ◻ n + , p ) for a pair of points x, x ′ ∈ ◻ n + distant from one another.
02 6. QUANTITATIVE CONVERGENCE OF THE SUBADDITIVE QUANTITIES
Lemma . There exists a constant C ∶= C ( d ) < ∞ such that for each n ∈ N and each p ∗ ∈ R d ×( d ) , (3.6) var µ β [ ∣ ◻ n ∣ ∑ x ∈◻ n ∇ v ( x, ⋅ , ◻ n + , p ∗ )] ≤ C −( d − ) n ∣ p ∗ ∣ . For later purposes, we also record that the variance of the flux contracts (3.7) var ⎡⎢⎢⎢⎢⎣ ∣ ◻ n ∣ ∑ x ∈◻ n ⎛⎝ ∇ v ( x, ⋅ , ◻ n + , p ∗ ) + β ∑ q ∈Q a q ∇ q v (⋅ , ⋅ , ◻ n + , p ∗ ) n q ( x )⎞⎠⎤⎥⎥⎥⎥⎦ ≤ C −( d − ) n ∣ p ∗ ∣ . Remark . The value of the coefficient d − is arbitrary; we can prove the result for any fixed numberstrictly smaller than d − β large enough accordingly. Remark . The argument presented in the proof below can be adapted to prove the variance estimate,for each point x ∈ ◻ n , var µ β [∇ v ( x, ⋅ , ◻ n + , p ∗ )] ≤ C. Since this estimate is an a priori estimate, we use it in Appendix B to prove the solvability of the Neumannproblem.
Proof.
We fix an inverse temperature β large enough so that all the regularity results of Chapter 5 holdwith the regularity exponent ε = . We decompose the argument into two steps. Step 1.
To ease the notation, we denote by v ∶= v (⋅ , ⋅ , ◻ n + , p ∗ ) . We assume without loss of generality that ∣ p ∗ ∣ =
1. We first decompose the variance(3.8) var µ β [ ∣ ◻ n ∣ ∑ x ∈◻ n ∇ v ( x, ⋅)] = ∣◻ n ∣ ∑ x,x ′ ∈◻ n cov µ β [∇ v ( x, ⋅) , ∇ v ( x ′ , ⋅)] . We then prove the estimate, for each pair of points x, x ′ ∈ ◻ n ,(3.9) ∣ cov µ β [∇ v ( x, ⋅) , ∇ v ( x ′ , ⋅)]∣ ≤ C n ∣ x − x ′ ∣ d − . The estimate (3.6) can then be deduced from (3.9) and (3.8); indeed we havevar µ β [ ∣ ◻ n ∣ ∑ x ∈◻ n ∇ v ( x, ⋅)] ≤ ∣◻ n ∣ ∑ x,x ′ ∈◻ n cov µ β [∇ v ( x, ⋅) , ∇ v ( x ′ , ⋅)]≤ C n ∣◻ n ∣ ∑ x,x ′ ∈◻ n ∣ x − x ′ ∣ d − ≤ C −( d − ) n . We now fix two points x, x ′ ∈ ◻ n and focus on the proof of (3.9). By applying the Helffer-Sj¨ostrand formula,we write(3.10) cov µ β [∇ v ( x, ⋅) , ∇ v ( x ′ , ⋅)] = ∑ y ∈ Z d ⟨ ∂ y ∇ v ( x, ⋅) H x ′ ( y, ⋅)⟩ µ β , where H z ′ is the solution of the Helffer-Sj¨ostrand equation, for each ( y, φ ) ∈ Z d × Ω, LH x ′ ( y, φ ) = ∂ y ∇ v ( x ′ , φ ) . We then decompose the function H x ′ according to the collection of Green’s matrices (G ∂ y ∇ v ( x ′ , ⋅) ) y ∈ Z d , followingthe notation introduced in (3.43) of Section 3.3 of Chapter 5. We obtain H x ′ ( y, φ ) = ∑ y ′ ∈ Z d G ∂ y ′ ∇ v ( x ′ , ⋅) ( y, φ ; y ′ ) . Using Proposition 4.7 of Chapter 3, we can estimate the L ( µ β ) -norm of the function H x ′ , for each point y ∈ Z d ,(3.11) ∥H x ′ ( y, ⋅)∥ L ( µ β ) ≤ C ∑ y ′ ∈ Z d ∥ ∂ y ′ ∇ v ( x ′ , ⋅)∥ L ( µ β ) ∣ y − y ′ ∣ d − . . QUANTITATIVE CONVERGENCE OF THE SUBADDITIVE QUANTITIES 103 We then claim that we have the estimates, for each pair of points y, y ′ ∈ Z d ,(3.12) ∥ ∂ y ∇ v ( x, ⋅)∥ L ( µ β ) ≤ C n ∣ y − x ∣ d + and ∥ ∂ y ′ ∇ v ( x ′ , ⋅)∥ L ( µ β ) ≤ C n ∣ y ′ − x ′ ∣ d + . The estimate (3.12) is proved in Step 2 below. Combining the inequalities (3.11), (3.12) and the formula (3.10),we obtain(3.13) cov µ β [∇ v ( x, ⋅) , ∇ v ( x ′ , ⋅)] ≤ C n ∑ y,y ′ ∈ Z d ∣ y ′ − x ′ ∣ d + × ∣ y − x ∣ d + × ∣ y − y ′ ∣ d − . The sum in the right side of the inequality (3.13) can be explicitly computed and we obtain the inequality (3.9).
Step 2. Proof of (3.12) . The argument relies on the differentiated Helffer-Sj¨ostrand equation introducedin Section 4 of Chapter 5 and on the reflection principle to solve the Neumann problem (3.16) below. Given acube Q ⊆ Z d of sidelength R , we recall the notation Q to denote the cube which has the same center as Q andsidelength R . We consider the specific cube ◻ ∶= ( , l n + ) d and the function v (⋅ , ⋅ , ◻ , p ∗ ) . Since the cube ◻ n + can be obtained from the cube ◻ by a translation and since the measure µ β is translation invariant, we see thatto prove the estimate (3.12) it is sufficient to prove the inequality, for each point y ∈ ◻ and each point z ∈ Z d ,(3.14) ∥ ∂ z ∇ v ( y, ⋅ , ◻ , p ∗ )∥ L ( µ β ) ≤ C n ∣ y − z ∣ d + . The reason justifying this specific choice of the cube ◻ will become clear later in the proof. Using the definitionof the map v ∶= v (⋅ , ⋅ , ◻ , p ∗ ) as a minimizer in the variational formulation of ν ∗ (◻ , p ∗ ) stated in (1.4), we seethat it is a solution of the Neumann problem(3.15) { ∆ φ v + L ◻ v = ◻ × Ω , n ⋅ ∇ v = n ⋅ p ∗ on ∂ ◻ × Ω , where the operator L ◻ is the uniformly elliptic operator defined by the formula L ◻ ∶= − β ∆ + β ∑ k ≥ β k ∇ k + ⋅ ( ◻ k ∇ k + ) + β ∇ ⋅ ( ◻∖◻ − ∇) + ∑ supp q ⊆◻ ∇ q ⋅ a q ∇ q , where we recall the notation ◻ k ∶= { x ∈ ◻ ∶ dist ( x, ∂ ◻) ≥ k } . The specific, technical formula of the operator L ◻ is not relevant in the proof; the important point of the argument is that the operator L ◻ is well-defined forfunctions which are only defined in the interior of the triadic cube ◻ and that, as it is the case for ellipticoperator L spat , it is uniformly elliptic and is a perturbation of the Laplacian − β ∆. As a consequence, allthe results stated in Chapter 5 for the Helffer-Sj¨ostrand operator L are also valid for the operator ∆ φ + L ◻ .In particular all the arguments stated in Section 4 about the differentiated Helffer-Sj¨ostrand equation applyin this setting. By applying the partial derivative ∂ to the system (3.15), we obtain that, if we denote by w ( y, z, φ ) = ∂ z v ( y, φ ) , then the function w is the solution of the system(3.16) ⎧⎪⎪⎪⎨⎪⎪⎪⎩ ∆ φ w + L ◻ ,y w + L spat ,z w = ∑ supp q ⊆◻ z ( β, q ) sin 2 π ( φ, q ) ( v, q ) q y ⊗ q z in ◻ × Z d × Ω , n ⋅ ∇ y w = ∂ ◻ × Z d × Ω , where the subscripts y (resp. z ) in the notation L ◻ ,y (resp. L spat ,z ) means that the spatial operator L ◻ n + (resp. L spat ,z ) only acts on the spatial variable y . We introduce the notation f to denote the function f ∶= ⎧⎪⎪⎪⎨⎪⎪⎪⎩ ◻ × ◻ × Ω → R d × d , ( y, z, φ ) ↦ ∑ supp q ⊆◻ z ( β, q ) cos 2 π ( φ, q ) ( v, q ) n q ( y ) ⊗ n q ( z ) . Using this notation, the system (3.16) becomes(3.17) ⎧⎪⎪⎨⎪⎪⎩ ∆ φ w + L ◻ ,y w + L spat ,z w = d y d z f in ◻ × Z d × Ω , n ⋅ ∇ y w = ∂ ◻ × Z d × Ω . To solve the system (3.17), we use the reflection principle. To this end, we need to introduce a few definitions,notations and remarks. We fix a point z ∈ Z d and extend the elliptic operator L ◻ , the functions v and f (⋅ , z ) ,initially defined on the cube ◻ , to the entire space according to a the following procedure. We let ̃◻ be
04 6. QUANTITATIVE CONVERGENCE OF THE SUBADDITIVE QUANTITIES the discrete cube (− l n + , l n + ) d . For each point x = ( x , . . . , x d ) ∈ ̃◻ , we let N x be the number of negativecoordinates of the components of x and define L ◻ ( x ) = L ◻ (∣ x ∣ , . . . , ∣ x d ∣) and f ( x, z, φ ) = (− ) N x f (∣ x ∣ , . . . , ∣ x d ∣ , z, φ ) . We extend the operator L ◻ and the function f periodically from the cube ̃◻ to Z d and let ̃ w be the solution ofthe elliptic system(3.18) ∆ φ ̃ w + L ◻ ,y ̃ w + L spat ,z ̃ w = d y d z ̃ f in Z d × Z d × Ω . It is straightforward to verify that with this construction, the restriction of the function ̃ w to the subcube ◻ satisfies the elliptic system (3.17); it is thus equal to the function w . We now study the function ̃ w . Wedenote by ̃G der the Green’s matrix associated to the operator ∆ φ + L ◻ ,y + L spat ,z . As was already mentioned,the operator L ◻ is a perturbation of the Laplacian β ∆; as a consequence, one can apply the same proofs asthe ones written in Chapter 5 and obtain the same results. In particular the statement of Proposition 4.2 ofChapter 5 holds for the differentiated Green’s matrix ̃G der . Using that the function ̃ w solves the system (3.18),we obtain the explicit formula ∇ y ̃ w ( y, z, φ ) = ∑ y ,z ∈ Z d ∇ y d ∗ y d ∗ z ̃G der , f ( y ,z , ⋅) ( y, z, φ ; y , z ) . Using the statement of Proposition 4.2, Chapter 5, we obtain the estimate on the L ( µ β ) -norm of the function ̃ w ,(3.19) ∥∇ y ̃ w ( y, z, ⋅)∥ L ( µ β ) ≤ C ∑ y ,z ∈ Z d ∥ f ( y , z , ⋅)∥ L ( µ β ) ∣ y − y ∣ d + + ∣ z − z ∣ d + . To compute (3.19), we prove the estimate, for each pair of points y , z ∈ Z d ,(3.20) ∥ f ( y , y + z , ⋅)∥ L ( µ β ) ≤ Ce − c √ β ∣ z ∣ ∑ y ∈ Z d e − c √ β ∣ y − y ∣ ∥∇ v ( y , ⋅)∥ L ( µ β ) . Let us make a comment about the estimate (3.20). Due to the exponential decay ∣ z ( β, q )∣ ≤ Ce − c √ β ∥ q ∥ , thefunction f decays exponentially fast outside the diagonal y = z of Z d . This phenomenon can be observedin the inequality (3.20): the exponential term e − c √ β ∣ z ∣ is small when the norm of z is large, i.e., whenthe point ( y , y + z ) is far from the diagonal {( y, y ) ∈ Z d × Z d } . Furthermore, on the diagonal, the term ∥ f ( y , y , ⋅)∥ L ( µ β ) is approximately equal to the value ∥∇ v ( y , ⋅)∥ L ( µ β ) ; but again the sum over all the chargesneeds to be taken into consideration and explains the sum over all the radii in the right side of (3.20) with theexponential decay e − c √ βr .We now prove the estimate (3.20). We start from the inequality, for each pair of points y , z ∈ Z d ,(3.21) ∥ f ( y , y + z , ⋅)∥ L ( µ β ) ≤ ∑ q ∈Q ∑ y ∈ supp n q e − c √ β ∥ q ∥ ∥∇ v ( y, ⋅)∥ L ( µ β ) ∥ n q ∥ L ∞ ∣ n q ( y )∣ ∣ n q ( y + z )∣ . We then note that if a charge q is such that the two points y and y + z belong to the support of n q , thenthe diameter of n q is larger than ∣ z ∣ and thus the diameter of q has to be larger than c ∣ z ∣ . From this remark,we deduce that(3.22) ∑ q ∈Q e − c √ β ∥ q ∥ ∥ n q ∥ L ∞ ∣ n q ( y )∣ ∣ n q ( y + z )∣ ≤ Ce − c √ β ∣ z ∣ . Similarly, if a charge q is such that the three points y and y + z and y belong to the support of n q , thenthe diameter of n q is larger than max (∣ z ∣ , ∣ y − y ∣) ≥ ∣ z ∣+∣ y − y ∣ . This implies that the diameter of q has to belarger than c (∣ z ∣ + ∣ y − y ∣) and we deduce that(3.23) ∑ q ∈Q e − c √ β ∥ q ∥ { y ∈ supp n q } ∥ n q ∥ L ∞ ∣ n q ( y )∣ ∣ n q ( y + z )∣ ≤ Ce − c √ β (∣ z ∣+∣ y − y ∣) . . QUANTITATIVE CONVERGENCE OF THE SUBADDITIVE QUANTITIES 105 Combining the estimates (3.21), (3.22) and (3.23), we obtain ∥ f ( y , y + z , ⋅)∥ L ( µ β ) ≤ ∑ q ∈Q ∑ y ∈ supp n q e − c √ β ∥ q ∥ ∥∇ v ( y , ⋅)∥ L ( µ β ) ∥ n q ∥ L ∞ n q ( y ) n q ( y + z )≤ ∑ y ∈ Z d ∑ q ∈Q e − c √ β ∥ q ∥ ∥∇ v ( y , ⋅)∥ L ( µ β ) { y ∈ supp n q } ∥ n q ∥ L ∞ n q ( y ) n q ( y + z )≤ Ce − c √ β ∣ z ∣ ∑ y ∈ Z d e − c √ β ∣ y − y ∣ ∥∇ v ( y , ⋅)∥ L ( µ β ) and we have proved the inequality (3.20).We now come back to the estimate (3.19), fix a point y ∈ ◻ and use the estimate (3.20). We obtain(3.24) ∥∇ y ̃ w ( y, z, φ )∥ L ( µ β ) ≤ C ∑ y ,y ,z ∈ Z d e − c √ β (∣ z − y ∣+∣ y − y ∣) ∥∇ v ( y , ⋅)∥ L ( µ β ) ∣ y − y ∣ d + + ∣ z − z ∣ d + . We focus on the sum over the variable y and z . The exponential decay of the terms e − c √ β ∣ z − y ∣ and e − c √ β ∣ y − y ∣ forces the sum to contract on the points y = y and z = y . We have the inequality, ∑ y ,z ∈ Z d e − c √ β (∣ z − y ∣+∣ y − y ∣) ∣ y − y ∣ d + + ∣ z − z ∣ d + ≤ C ∣ y − y ∣ d + + ∣ y − z ∣ d + . Using the previous estimate, we can simplify the inequality (3.24) and we obtain ∥∇ y ̃ w ( y, z, φ )∥ L ( µ β ) ≤ C ∑ y ∈ Z d ∥∇ v ( y , ⋅)∥ L ( µ β ) ∣ y − y ∣ d + + ∣ y − z ∣ d + . We then truncate the sum, depending on whether the point y belongs to the cube ◻ . We write(3.25) ∥∇ y ̃ w ( y, z, φ )∥ L ( µ β ) ≤ C ∑ y ∈ ◻ ∥∇ v ( y , ⋅)∥ L ( µ β ) ∣ y − y ∣ d + + ∣ y − z ∣ d + ·„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„¶ (3.25) −( i ) + C ∑ y ∈ Z d ∖ ◻ ∥∇ v ( y , ⋅)∥ L ( µ β ) ∣ y − y ∣ d + + ∣ y − z ∣ d + ·„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„¶ (3.25) −( ii ) . We treat the two terms in the right side of (3.25) separately. For the term (3.25)-(i), we use that the map v isa solution of the Helffer-Sj¨ostrand equation (3.15) in the cube ◻ and apply Proposition 2.4 of Chapter 5 withthe regularity exponent ε = . We obtain, for each point y ∈ ◻ ,(3.26) ∥∇ v ( y , ⋅)∥ L ( µ β ) ≤ C ( l n + ) ∥∇ v ∥ L (◻ ,µ β ) ≤ C n , where we used Remark 1.7 and the inequality (1.7) in the second inequality. Using the estimate (3.26), we cancompute the term (3.25)-(i) ∑ y ∈ ◻ ∥∇ v ( y , ⋅)∥ L ( µ β ) ∣ y − y ∣ d + + ∣ y − z ∣ d + ≤ C εn ∑ y ∈ ◻ ∣ y − y ∣ d + + ∣ y − z ∣ d + (3.27) ≤ C εn ∑ y ∈ Z d ∣ y − y ∣ d + + ∣ y − z ∣ d + ≤ C εn ∑ y ∈ Z d ∣ y ∣ d + + ∣ y + y − z ∣ d + ≤ C εn ∣ y − z ∣ d + , where we used Proposition 0.5 of Appendix C in the last inequality. We now treat the term (3.25)-(ii). In thatcase, we use the estimate ∣ y − y ∣ ≥ c ∣ y ∣ valid for any point y ∈ Z d ∖ ◻ and any point y ∈ ◻ . We obtain theinequality ∑ y ∈ Z d ∖ ◻ ∥∇ v ( y , ⋅)∥ L ( µ β ) ∣ y − y ∣ d + + ∣ y − z ∣ d + ≤ ∑ y ∈ Z d ∖ ◻ ∥∇ v ( y , ⋅)∥ L ( µ β ) ∣ y ∣ d + + ∣ y − z ∣ d + .
06 6. QUANTITATIVE CONVERGENCE OF THE SUBADDITIVE QUANTITIES
To estimate the previous inequality, we partition the space into cubes. We consider the set
K ∶= l n + Z d andnote that the collection of cubes ( κ + ◻) κ ∈K is a partition of Z d . We note that, for each point κ ∈ K ∖ { } ,each point y ∈ ( κ + ◻) and each point z ∈ Z d , one has the inequalities(3.28) c (∣ y ∣ d + + ∣ y − z ∣ d + ) ≤ ∣ κ ∣ d + + ∣ κ − z ∣ d + ≤ C (∣ y ∣ d + + ∣ y − z ∣ d + ) . Using the extension of the function v from the cube ◻ to the entire space Z d stated in Step 2 of the proof ofLemma 3.2, the volume identity ∣ ◻∣ = ( ) d ∣◻∣ and the inequality (1.7), we obtain the estimate, for each κ ∈ K ,(3.29) ∥ v ∥ L ( κ + ◻ ,µ β ) ≤ C. A combination of the inequalities (3.28) and (3.29) yields ∑ y ∈ Z d ∖ ◻ ∥∇ v ( y , ⋅)∥ L ( µ β ) ∣ y ∣ d + + ∣ y − z ∣ d + ≤ ∑ κ ∈K∖{ } ∑ y ∈ κ + ◻ ∥∇ v ( y , ⋅)∥ L ( µ β ) ∣ y ∣ d + + ∣ y − z ∣ d + (3.30) ≤ ∑ κ ∈K∖{ } ∑ y ∈ κ + ◻ ∥∇ v ( y , ⋅)∥ L ( µ β ) ∣ κ ∣ d + + ∣ κ − z ∣ d + ≤ C ∑ κ ∈K∖{ } ∣ ◻∣∣ κ ∣ d + + ∣ κ − z ∣ d + . To estimate the sum in the right side of (3.30), we use the estimate (3.28) a second time and write ∑ y ∈ Z d ∖ ◻ ∥∇ v ( y , ⋅)∥ L ( µ β ) ∣ y ∣ d + + ∣ y − z ∣ d + ≤ C ∑ κ ∈K∖{ } ∣ ◻∣∣ κ ∣ d + + ∣ κ − z ∣ d + (3.31) ≤ C ∑ κ ∈K∖{ } ∑ y ∈ κ + ◻ ∣ κ ∣ d + + ∣ κ − z ∣ d + ≤ C ∑ κ ∈K∖{ } ∑ y ∈ κ + ◻ ∣ y ∣ d + + ∣ y − z ∣ d + ≤ C ∑ y ∈ Z d ∖ ◻ ∣ y ∣ d + + ∣ y − z ∣ d + ≤ C max (∣ z ∣ , n ) d + . where we used Remark 0.6 of Appendix C in the last inequality. We finally slightly modify the result: usingthat the point y belongs to the cube ◻ , we have the inequality(3.32) 1 (∣ z ∣ ∨ n ) d + ≤ C ∣ z − y ∣ d + . We use the computation (3.31) and the inequality (3.32) to obtain(3.33) ∑ y ∈ Z d ∖ ◻ ∥∇ v ( y , ⋅)∥ L ( µ β ) ∣ y ∣ d + + ∣ y − z ∣ d + ≤ C ∣ z − y ∣ d + . By combining the estimates (3.25), (3.27) and (3.33), we deduce that(3.34) ∥∇ y ̃ w ( y, z, ⋅)∥ L ( µ β ) ≤ C n ∣ z − y ∣ d + . We complete the argument by recalling that for each y ∈ ◻ and each z ∈ Z d , the function ̃ w is defined so thatwe have ∇ y ̃ w ( y, z, ⋅) = ∂ z ∇ v ( y, ⋅ , ◻) . The inequality (3.34) can thus be rewritten ∥ ∂ z ∇ v ( y, ⋅ , ◻ , p ∗ )∥ ≤ C n ∣ y − z ∣ d + . The proof of the inequality (3.14) and thus of Step 2 is complete. (cid:3) . QUANTITATIVE CONVERGENCE OF THE SUBADDITIVE QUANTITIES 107 L -norms of the functions u − l p and v − a ∗ (◻ n ) − l p ∗ . The objective of thissection is to prove that the optimizers u and v are close in the L (◻ n , µ β ) -norm to affine functions. Theresult relies on the multiscale Poincar´e inequality stated in Appendix A and is quantified in terms of thesubadditivity defect τ n . Lemma L estimate for the optimizers u and v ) . There exist an inverse temperature β ∶= β ( d ) < ∞ and a constant C ∶= C ( d ) < ∞ such that for each β > β , each integer n ∈ N , and each pair of vectors p, p ∗ ∈ R d ×( d ) , (3.35) ∥ u (⋅ , ⋅ , ◻ − n + , p ) − l p ∥ L (◻ n + ,µ β ) ≤ C ∣ p ∣ n ( − n + n ∑ m = − m − n τ m ) , and (3.36) ∥ v (⋅ , ⋅ , ◻ n + , p ∗ ) − l a − n p ∗ − ( v ) ◻ n + ,µ β ∥ L (◻ n + ,µ β ) ≤ C ∣ p ∗ ∣ n ( − n + n + ∑ m = − ( m − n ) τ m ) . Proof.
We assume without loss of generality that ∣ p ∣ = ∣ p ∗ ∣ =
1. The strategy of the proof relies ontwo ingredients: ● First, we need to estimate the spatial averages of the gradients of the functions u − l p and v − l a ∗ (◻ n ) − p ∗ and prove that they are small. To be more precise, we estimate these spatial averages in terms of thesubadditivity defects τ n . The proof relies on different arguments depending on which function weconsider: – For the function u associated to the energy quantity ν , we use the subadditivity property statedin Proposition 2.1 and the following fact: for any discrete cube ◻ ⊆ Z d and any function f ∶ ◻ → R which is equal to 0 on the boundary of the cube ◻ , one has the identity ∑ x ∈◻ ∇ f ( x ) = – For the function v associated to the energy quantity ν ∗ , we use the subadditivity property statedin Proposition 2.1 and Lemma 3.2 to control the variance of the spatial average of its gradients. ● The multiscale Poincar´e inequality, which is stated in Proposition 0.1 in Appendix A. This inequalityallows to estimate the L -norm of a function in terms of the spatial averages of its gradient.We first focus on the function u ∶= u (⋅ , ⋅ , ◻ − n + , p ) and prove the inequality (3.35). We first recall that thefunction u is extended by the affine function l p outside the cube ◻ − n + . We thus have ∥ u (⋅ , ⋅ , ◻ − n + , p ) − l p ∥ L (◻ n + ,µ β ) = ∥ u (⋅ , ⋅ , ◻ − n + , p ) − l p ∥ L (◻ − n + ,µ β ) . By the multiscale Poincar´e inequality stated in Proposition 0.1 of Appendix A, we have(3.37) ∥ u (⋅ , ⋅ , ◻ − n + , p ) − l p ∥ L (◻ − n + ,µ β ) ≤ C ∥∇ u (⋅ , ⋅ , ◻ − n + , p ) − p ∥ L (◻ n + ,µ β ) ·„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„¶ (3.37) −( i ) + C n n ∑ m = m ∣Z m,n ∣ ⟨( ∣ z + ◻ m ∣ ∑ x ∈ z +◻ m ∇ u (⋅ , ⋅ , ◻ − n + , p ) − p ) ⟩ µ β ·„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„¶ (3.37) −( ii ) . We bound the first term (3.37)-(i) using the estimate (1.7). We obtain the inequality(3.38) ∥∇ u (⋅ , ⋅ , ◻ − n + , p ) − p ∥ L (◻ n + ,µ β ) ≤ ∥∇ u (⋅ , ⋅ , ◻ n + , p )∥ L (◻ − n + ,µ β ) + ∣ p ∣ ≤ C ∣ p ∣ . To estimate the term (3.37)-(ii), we use the two following ingredients: ● The subadditivity of the energy ν which is stated in Proposition 2.1 and Remark 2.2. It reads, foreach integer m ∈ { , . . . , n } , ∣Z m,n ∣ − ∑ z ∈Z m,n (cid:74) u (⋅ , ⋅ , ◻ − n + , p ) − u (⋅ , ⋅ , z + ◻ − m , p ) (cid:75) H ( z +◻ − m ,µ β ) ≤ C ( ν (◻ − m , p ) − ν (◻ − n + , p ) + − m ∣ p ∣ )≤ C ( n ∑ k = m τ k + − m ∣ p ∣ ) .
08 6. QUANTITATIVE CONVERGENCE OF THE SUBADDITIVE QUANTITIES ● For each point z ∈ Z m,n , the function u (⋅ , z + ◻ m , p ) belongs to the space l p + H ( z + ◻ m , µ β ) . Thisimplies that, for each realization of the field φ ∈ Ω,(3.39) 1 ∣ z + ◻ − m ∣ ∑ x ∈ z +◻ m ∇ u ( x, φ, z + ◻ − m , p ) = p. We deduce the inequality, for each integer m ∈ { , . . . , n } ,(3.40) ∑ z ∈Z m,n ∣Z m,n ∣ ⟨( ∣ z + ◻ m ∣ ∑ x ∈ z +◻ m ∇ u ( x, ⋅ , ◻ − n + , p ) − p ) ⟩ µ β ≤ C ( n ∑ k = m τ k + − m ∣ p ∣ ) . Combining the estimates (3.37), (3.38) and (3.40) completes the proof of the estimate (3.35).We now prove the inequality (3.36). By the multiscale Poincar´e inequality, we have(3.41) ∥ v (⋅ , ⋅ , ◻ n + , p ∗ ) − l a − n p ∗ − ( v (⋅ , ⋅ , ◻ n + , p ∗ ) − l a − n p ∗ ) ◻ n + ∥ L (◻ n + ,µ β ) ≤ C ∥∇ v (⋅ , ◻ n + , p ∗ ) − a − n p ∗ ∥ L (◻ n + ,µ β ) ·„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„¶ (3.41) −( i ) + C n n ∑ m = m ∣Z m,n ∣ ⟨( ∣ z + ◻ m ∣ ∑ x ∈ z +◻ m ∇ v (⋅ , ◻ n + , p ∗ ) − a − n p ∗ ) ⟩ µ β ·„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„¶ (3.41) −( ii ) . We first treat the term on the left side. Since the average value of a linear map on a cube centered at 0 isequal to 0, we have that ( v (⋅ , ⋅ , ◻ n + , p ∗ ) − l a − n p ∗ ) ◻ n + = ∣◻ n + ∣ ∑ x ∈◻ n + v ( x, ⋅ , ◻ n + , p ∗ ) − l a − n p ∗ ( x ) = ∣◻ n + ∣ ∑ x ∈◻ n + v ( x, ⋅ , ◻ n + , p ∗ ) . We then use the estimate (1.6) and the inclusion ◻ n + ⊆ ◻ n + . We obtain ∥( v (⋅ , ⋅ , ◻ n + , p ∗ ) − l a − n p ∗ ) ◻ − ( v (⋅ , ⋅ , ◻ n + , p ∗ )) ◻ n + ,µ β ∥ L ( µ β ) = var µ β [( v (⋅ , ⋅ , ◻ n + , p ∗ )) ◻ n + ] (3.42) ≤ C ∣◻ n ∣ ∑ x ∈◻ n + var µ β [ v ( x, ⋅ , ◻ n + , p ∗ )]≤ C ∣◻ n ∣ ∑ x ∈ ◻ n + var [ v ( x, ⋅ , ◻ n + , p ∗ )]≤ C ∣ p ∗ ∣ . The first term (3.41)-(i) can be estimated with the same argument as in the inequality (3.38). We obtain(3.43) ∥∇ v (⋅ , ⋅ , ◻ n + , p ∗ ) − a − n p ∗ ∥ L (◻ n + ,µ β ) ≤ C. To estimate the term (3.41)-(ii), we prove that, for each integer m ∈ { , . . . , n } ,(3.44) 1 ∣Z m,n ∣ ⟨( ∣ z + ◻ m ∣ ∑ x ∈ z +◻ m ∇ v (⋅ , ⋅ , ◻ n + , p ∗ ) − a − n p ∗ ) ⟩ µ β ≤ C − m + C n ∑ k = m τ k . To this end, we decompose the left side of (3.44) and write1 ∣Z m,n ∣ ∑ z ∈Z m,n ⟨( ∣ z + ◻ m ∣ ∑ x ∈ z +◻ m ∇ v (⋅ , ⋅ , ◻ n + , p ∗ ) − a − n p ∗ ) ⟩ µ β (3.45) ≤ ∣Z m,n ∣ − ∑ z ∈Z m,n (cid:74) v (⋅ , ⋅ , ◻ n + , p ∗ ) − v (⋅ , ⋅ , z + ◻ m , p ∗ ) (cid:75) H ( z +◻ m ,µ β ) + ∣ a − n p ∗ − a − m p ∗ ∣ + ∣Z m,n ∣ − ∑ z ∈Z m,n ⟨( ∣ z + ◻ m ∣ ∑ x ∈ z +◻ m ∇ v (⋅ , ⋅ , z + ◻ m , p ∗ ) − a − m p ∗ ) ⟩ µ β . . QUANTITATIVE CONVERGENCE OF THE SUBADDITIVE QUANTITIES 109 We estimate the first term on the right side by Proposition 2.5, the second term by Lemma 3.1. We obtain(3.46) ∣Z m,n ∣ − ∑ z ∈Z m,n (cid:74) v (⋅ , ⋅ , ◻ n + , p ∗ ) − v (⋅ , ⋅ , z + ◻ m , p ∗ ) (cid:75) H ( z +◻ m ,µ β ) + ∣ a − n p ∗ − a − m p ∗ ∣ ≤ C − m + C n ∑ k = m τ k . There remains to estimate the third term in the right side of (3.45). We first recall the identity, for eachinteger m ∈ N , 1 ∣ ◻ m ∣ ∑ x ∈◻ m ⟨∇ v ( x, ⋅ , ◻ m , p ∗ )⟩ µ β = a − m p ∗ . We use the translation invariance of the measure µ β and Lemma 3.2. To ease the notation, we note that indimension larger than 3, we have the estimate d − ≥ . We obtain ∣Z m,n ∣ − ∑ z ∈Z m,n ⟨( ∣ z + ◻ m ∣ ∑ x ∈ z +◻ m ∇ v ( x, ⋅ , z + ◻ m , p ∗ ) − a − m p ∗ ) ⟩ µ β = ⟨( ∣◻ m ∣ ∑ x ∈◻ m ∇ v ( x, ⋅ , ◻ m , p ∗ ) − a − m p ∗ ) ⟩ µ β (3.47) = var µ β [ ∣ ◻ m ∣ ∑ x ∈◻ n ∇ v ( x, ⋅ , ◻ m , p ∗ )]≤ C ( − m + τ m ) . Combining the estimates (3.42), (3.43), (3.45), (3.47) and (3.46) completes the proof of (3.36). (cid:3) J . In this section, we deduce from the previous results and the Caccioppoliinequality a control over the energy quantity J (◻ n , p, a n p ) . The argument needs to take into account theinfinite range of the Helffer-Sj¨ostrand operator and the specific forms of the energies E and E ∗ which causessome technicalities in the argument. The result is stated in the lemma below. Lemma . There exist an inverse temperature β ∶= β ( d ) < ∞ and a constant C ∶= C ( d ) < ∞ such thatfor each β ≥ β , each integer n ∈ N and each p ∈ R d ×( d ) , (3.48) ν (◻ − n , p ) + ν ∗ (◻ n , a n p ) − a n ∣ p ∣ ≤ C ∣ p ∣ ( − n + n + ∑ m = − n − m τ m ) . Proof.
The strategy of the proof relies on three ingredients: the Caccioppoli inequality stated inProposition 1.1 of Chapter 5, the one-sided convex duality formula (1.21) stated in Proposition 1.12 and the L -norm estimate on the optimizers u and v stated in Lemma 3.5.We fix a slope p ∈ R d and assume without loss of generality that ∣ p ∣ =
1. By Proposition 1.12, we have theidentity ν (◻ − n , p ) + ν ∗ (◻ n , a n p ) − a n ∣ p ∣ = E ∗◻ n [ u (⋅ , ⋅ , ◻ − n , p ) − v (⋅ , ⋅ , ◻ n , a n p )] + O ( C − m ) . To prove the estimate (3.48), it is thus sufficient to prove the estimate(3.49) E ∗◻ n [ u (⋅ , ⋅ , ◻ − n , p ) − v (⋅ , ⋅ , ◻ n , a n p )] ≤ C ( − n + n + ∑ m = − n − m τ m ) . Using the coercivity of the energy E ∗◻ n stated in (1.2), we see that to prove the inequality (3.49), it is sufficientto prove the estimate(3.50) (cid:74) u (⋅ , ⋅ , ◻ − n , p ) − v (⋅ , ⋅ , ◻ n , a n p ) (cid:75) H (◻ n ,µ β ) ≤ C ( − n + n + ∑ m = − n − m τ m ) , and by Propositions 2.1 and 2.5, we see that to prove (3.50) it is sufficient to prove(3.51) (cid:74) u (⋅ , ⋅ , ◻ − n + , p ) − v (⋅ , ⋅ , ◻ n + , a n p ) (cid:75) H (◻ n ,µ β ) ≤ C ( − n + n + ∑ m = − n − m τ m ) . We now focus on the proof of (3.51). In the rest of the proof, we make use of the notations u ∶= u (⋅ , ⋅ , ◻ − n + , p ) and v ∶= v (⋅ , ⋅ , ◻ n + , a n p ) − ( v (⋅ , ⋅ , ◻ n + , a n p )) ◻ n + ,µ β . By Lemma 3.5, we have the L (◻ n + , µ β ) -estimate(3.52) ∥ u − v ∥ L (◻ n + ,µ β ) ≤ ∥ u − l p ∥ L (◻ n + ,µ β ) + ∥ v − l p ∥ L (◻ n + ,µ β ) ≤ C n ( − n + n + ∑ m = − m − n τ m ) .
10 6. QUANTITATIVE CONVERGENCE OF THE SUBADDITIVE QUANTITIES
We recall the following notation: for each integer k ∈ N , we denote by ◻ kn + the interior cube ◻ kn + ∶={ x ∈ ◻ n + ∶ dist ( x, ∂ ◻ n + ) ≥ k } . By the first variation formula stated in Proposition 1.12, the maps u and v are solutions of the equations L u = ◻ − n + × Ω and L ◻ n + , ∗ v = ◻ n + × Ω , where we recall the definition of the Helffer-Sj¨ostrand operator L ◻ n + , ∗ L ◻ n + , ∗ ∶= ∆ φ − β ∆ + β ∑ k ≥ (− ) k + β k ∇ k + ⋅ ( ◻ kn + ∇ k + ) − β ∇ ⋅ ( ◻∖◻ − ∇) + ∑ supp q ⊆◻ n + ∇ q ⋅ a q ∇ q . One can adapt the proof of the Caccioppoli inequality (Proposition 1.1 of Chapter 5) to the operator L ◻ n + , ∗ and obtain the following statement. There exists a constant C ∶= C ( d ) < ∞ such that for any vectorfields F ∶ ◻ n + × Ω → R d ×( d ) and G ∶ ◻ n + × Ω → R d , any ball B ( x, r ) such that B ( x, r ) is included in the cube ◻ n + and every solution w ∶ ◻ n + × Ω → R ( d ) of the equation L ◻ n + , ∗ w = ∇ ⋅ F + d G in B ( x, r ) × Ω , one has the estimate(3.53) (cid:74) w (cid:75) H ( B r ( x ) ,µ β ) ≤ CR ∥ w ∥ L ( B r ( x ) ,µ β ) + ∥ F ∥ L ( B r ( x ) ,µ β ) + ∥ G ∥ L ( B r ( x ) ,µ β ) + ∑ y ∈◻ n + ∖ B r ( x ) e − c ( ln β )∣ y − x ∣ ∥ w ( y, ⋅)∥ L ( µ β ) . We then note that, by the definition of the operator L ◻ n + , ∗ , the function u satisfies the equation L ◻ n + , ∗ u = ∇ ⋅ F + d G in ◻ − n + × Ω , where the vector fields F and G are defined by the formulas F ∶= − β ∑ k ≥ dist ( x,∂ ◻ n + ) β k (− ∆ ) k ∇ u and G ∶= ∑ supp q /⊆◻ n + a q (∇ q u ) n q . We estimate the L (◻ − n + , µ β ) -norm of the functions F and G . We first note that every point x in the cube ◻ − n + satisfies the inequality dist ( x, ∂ ◻ n + ) ≥ c n . Using the boundedness of the discrete Laplacian operator,the upper bound on the L -norm of the gradient of the function u stated in (1.7) and choosing the inversetemperature β large enough, we have ∥ F ∥ L (◻ − n + ,µ β ) = XXXXXXXXXXXX ∑ k ≥ dist ( x,∂ ◻ n + ) β k (− ∆ ) k ∇ u XXXXXXXXXXXX L (◻ − n + ,µ β ) (3.54) ≤ ∑ k ≥ dist ( x,∂ ◻ n + ) β k ∥ ∆ k ∇ u ∥ L (◻ − n + ,µ β ) ≤ ∑ k ≥ dist ( x,∂ ◻ n + ) C k β k ∥∇ u ∥ L (◻ − n + ,µ β ) ≤ − Cβ ( Cβ ) c n ∥∇ u ∥ L (◻ − n + ,µ β ) ≤ Ce − c ( ln β ) n . Using a similar argument, we note that for each point x in the interior cube ◻ − n + , if a charge q ∈ Q is suchthat its support is not included in the cube ◻ n + and such that the point x belongs to the support of n q , thenits diameter must be larger than c n . We can then use the estimate on the coefficient a q stated in (1.9) andthe estimate (1.7) to obtain(3.55) ∥ G ∥ L (◻ − n + ,µ β ) = XXXXXXXXXXX ∑ supp q /⊆◻ n + a q (∇ q u ) n q XXXXXXXXXXX L (◻ − n + ,µ β ) ≤ Ce − c √ β n ∥∇ u ∥ L (◻ − n + ,µ β ) ≤ Ce − c √ β n . . QUANTITATIVE CONVERGENCE OF THE SUBADDITIVE QUANTITIES 111 We now apply the Caccioppoli inequality (1.2), Chapter 5, to the function w ∶= u − v which is solution of theequation L ◻ n + , ∗ ( u − v ) = ∇ ⋅ F + d G in the set ◻ − n + × Ω. We obtain β ∑ y ∈ Z d ∥ ∂ y ( u − v )∥ L (◻ n ,µ β ) + ∥∇ ( u − v )∥ L (◻ n ,µ β ) (3.56) ≤ C − n ∥ u − v ∥ L (◻ − n + ,µ β ) ·„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„¶ (3.56) −( i ) + ∥ F ∥ L (◻ − n + ,µ β ) + ∥ G ∥ L (◻ − n + ,µ β ) ·„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„¶ (3.56) −( ii ) + ⎛⎝ ∑ x ∈◻ − n + ∖◻ n e − c ( ln β )∣ x ∣ ∥ u ( x, ⋅) − v ( x, ⋅)∥ L ( µ β ) ⎞⎠ ·„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„¶ (3.56) −( iii ) . We estimate the term (3.56)-(i) thanks to the inequality (3.52). We obtain(3.57) C − n ∥ u − v ∥ L (◻ − n + ,µ β ) ≤ C ( − n + n + ∑ m = − m − n τ m ) . We estimate the term (3.56)-(ii) by the inequalities (3.54) and (3.55). We obtain(3.58) ∥ F ∥ L (◻ − n + ,µ β ) + ∥ G ∥ L (◻ − n + ,µ β ) ≤ Ce − c ( ln β ) n . For the term (3.56)-(iii), we use the estimate (3.52), the observation τ n ≤ C and note that if a point x liesoutside the cube ◻ n then its norm must be larger than c n . We obtain ∑ x ∈◻ n + ∖◻ n e − c √ β ∣ x ∣ ∥( u − v )( x, ⋅)∥ L ( µ β ) ≤ Ce − c √ β n ∑ x ∈◻ n + ∥∇( u − v )( x, ⋅)∥ L ( µ β ) (3.59) ≤ Ce − c √ β n dn ∥ u − v ∥ L (◻ n + ,µ β ) ≤ Ce − c √ β n n ( d + ) ≤ Ce − c √ β n . Combining the estimates (3.56), (3.57), (3.58) and (3.59) completes the proof of Lemma 3.6. (cid:3) J . In this section, we use Lemma 3.6 togetherwith an iteration argument to obtain an algebraic rate of convergence for the quantity J (◻ n , p, a n p ) . Thestrategy implemented in the proof is essentially the one described in the paragraph following Proposition 1.10 upto a technical difficulty: the term in the right side of the estimate (3.48) of Lemma 3.6 is not the subadditivitydefect τ n but a weighted average the subadditivity defects. This additional technicality requires to make useof a weighted quantity denoted by ̃ F n in the proof below. Proposition . There exist a constant C ∶= C ( d ) < ∞ and an exponent α ∶= α ( d ) > such that for eachinteger n ∈ N and each p ∈ R d ×( d ) , ν (◻ − n , p ) + ν ∗ (◻ n , a n p ) − a n ∣ p ∣ ≤ C ∣ p ∣ − αn . We record, as a corollary, that the quantitative rate of convergence established in Proposition 3.7 impliesa quantitative estimate on the subadditivity defect τ n . The result is stated below. Corollary . There exist a constant C ∶= C ( d ) < ∞ and an exponent α ∶= α ( d ) > such that for eachinteger n ∈ N , (3.60) − C − n ≤ τ n ≤ C − αn . Proof of Proposition 3.7 and Corollary 3.8.
For k ∈ N , we let B ( R k ) be the unit ball in R k . Wedenote by C the constant which appears in the right side of the identity (1.21) and define, for each integer n ∈ N , F n ∶= sup p ∈ B ( R d ×( d ) ) ν (◻ − n , p ) + ν ∗ (◻ n , a n p ) − a n ∣ p ∣ + C ∣ p ∣ − n .
12 6. QUANTITATIVE CONVERGENCE OF THE SUBADDITIVE QUANTITIES
We note that by the inequality (1.5), we have the upper bound, for each integer n ∈ N , F n ≤ C . ByProposition 1.12 and Lemma 3.6, we have for each integer n ∈ N ,(3.61) 0 ≤ F n ≤ C ( − n + n + ∑ m = − n − m τ m ) . Additionally, we obtain from the subadditivity properties stated in Propositions 2.1 and 2.5 the inequality(3.62) F n + ≤ F n + C − n . Combining the estimates (3.61) and (3.62) implies that0 ≤ F n + ≤ C ( − n + + n + ∑ m = − n − m τ m ) . By definition of the subadditivity defect τ n , and the fact that the maps p → ν (◻ n , p ) − ν (◻ n + , p ) + C ∣ p ∣ − n and p ∗ → ν (◻ n , p ∗ ) − ν (◻ n + , p ∗ ) + C ∣ p ∗ ∣ − n are quadratic and non-negative, we have τ n ≤ C d ∑ k = ( ν (◻ n , e k ) − ν (◻ n + , e k ) + ν ∗ (◻ n , e k ) − ν ∗ (◻ n + , e k )) + C − n (3.63) ≤ C ( F n − F n + + − n ) . We then define ̃ F n ∶= − n ∑ nk = k F k . From the estimates (3.61) and (3.63) and the inequality F ≤ C , wededuce ̃ F n − ̃ F n + = − n n ∑ k = k ( F k − F k + ) − − ( n + ) F ≥ − n n ∑ k = k ( C τ k − − k ) − C − n (3.64) ≥ C n ∑ k = − ( n − k ) τ k − n ∑ k = − ( n − k ) − k − C − n ≥ C n ∑ k = − ( n − k ) τ k − C − n . We then compute by using the inequalities (3.62) and (3.61) ̃ F n + = − n + k + ∑ k = k F k = − n n ∑ k = k F k + + − n + F ≤ − n n ∑ k = k ( F k + C − n ) + C − n ≤ ̃ F n + C − n . We then use the estimate (3.61) and write ̃ F n + ≤ − n n ∑ k = k F k + C − n ≤ − n n ∑ k = k ( C − k + k ∑ m = − k − m τ m ) + C − n (3.65) ≤ C − n n ∑ k = − k + − n n ∑ k = − k k ∑ m = − m τ m + C − n ≤ C n ∑ k = − n − k τ k + C − n . By combining the estimates (3.64) and (3.65), we have obtained ̃ F n + ≤ C ( ̃ F n − ̃ F n + ) + C − n . The previous inequality can be rewritten(3.66) ̃ F n + ≤ CC + ̃ F n + C − n . We set α ∶= ln CC + so that we have 3 α = CC + and define the exponent α ∶= min ( α , ) . We iterate theinequality (3.66) and note that the inequality F ≤ C implies the inequality ̃ F ≤ C . We obtain ̃ F n ≤ − α n ̃ F + C n ∑ k = − α k − n − k ≤ C − αn . . DEFINITION OF THE FIRST-ORDER CORRECTOR AND QUANTITATIVE SUBLINEARITY 113 Finally, by the definition of the weighted sum ̃ F n , we have the inequality F n ≤ ̃ F n . The proof of Proposition 3.7is complete.There only remains to prove Corollary 3.8. The lower bound in (3.60) is a direct consequence subadditivityproperties stated in Propositions 2.1 and 2.5. For the upper bound, we use the inequality (3.63) together withthe estimates F n ≤ C − αn and F n + ≥ (cid:3) ν and ν ∗ . In this section,we deduce Proposition 1.10 from Proposition 3.7.
Proof of Proposition 1.10.
Before starting the proof, we collect some ingredients which were provedin this chapter: ● By Proposition 1.12 and Definition 3.1, we have the identities, for each integer n ∈ N and each p, p ∗ ∈ R d ,(3.67) ν (◻ − n , p ) = p ⋅ a (◻ − n ) p and ν ∗ (◻ n , p ∗ ) = p ∗ ⋅ a − n p ∗ ; ● By Property (4) of Proposition 1.12, there exist two strictly positive constants c, C depending onlyon the dimension d such that, for every cube ◻ ⊆ Z d ,(3.68) c ≤ a (◻) , a ∗ (◻) ≤ C ; ● By Corollaries 2.3 and 2.6, we have the convergences(3.69) a (◻ − n ) —→ n →∞ a and a − n —→ n →∞ a − ∗ ; ● By the one sided convex duality estimate (1.21) and Proposition 3.7, we have the inequalities, foreach p ∈ R d ×( d ) , − C ∣ p ∣ − n ≤ ν (◻ n , p ) + ν ∗ (◻ n , a n p ) − a n ∣ p ∣ ≤ C ∣ p ∣ − αn , which can be rewritten, by using (3.67),(3.70) − C − n ≤ a (◻ − n ) − a n ≤ C − αn ; ● By Lemma 3.1 and Corollary 3.8, we have the inequality, for each pair of integers ( m, n ) ∈ N suchthat m ≤ n ,(3.71) ∣ a − n − a − m ∣ ≤ n ∑ k = m τ k + C − m ≤ n ∑ k = m C − αk + C − m ≤ C − αm . We now combine the four previous results to complete the proof of Proposition 1.10. First by sending n toinfinity in the inequality (3.70) and using the convergence (3.69), we obtain the identity a = a − ∗ . Then bysending n to infinity in the inequality (3.71), we obtain the inequality, for each integer m ∈ N ,(3.72) ∣ a − m − a − ∣ ≤ C − αm . We then combine the inequality (3.68) with the inequality (3.72) to obtain(3.73) ∣ a m − a ∣ ≤ C − αm . Combining the estimate (3.73) with the estimate (3.70) and using that the exponent α is smaller than , wededuce that, for each integer n ∈ N ,(3.74) ∣ a (◻ − n ) − a ∣ ≤ ∣ a (◻ − n ) − a n ∣ + ∣ a n − a ∣ ≤ C − αn . Proposition 1.12 is then a consequence of the estimates (3.73), (3.74) and the representation formulas (3.67). (cid:3)
4. Definition of the first-order corrector and quantitative sublinearity
An important ingredient to prove the quantitative homogenization of the mixed derivative of the Green’smatrix associated to the Helffer-Sj¨ostrand operator L (which is the subject of Section 7) is the first-ordercorrector. The objective of this section is to introduce this function and to deduce from the algebraic rate ofconvergence on the energy ν established in Proposition 1.10 two properties on this map: ● The quantitative sublinearity of the corrector, this is stated in the equation (4.1); ● A quantitative estimate on the H − -norm of the flux of the corrector, this is stated in the estimate (4.2).
14 6. QUANTITATIVE CONVERGENCE OF THE SUBADDITIVE QUANTITIES
The corrector which is introduced in this section is a finite-volume version of the corrector (see Definition 4.1),the reason justifying this choice is that it is is simpler to construct from the subadditive energy ν thanthe infinite-volume corrector and allows the arguments developed in Chapter 7 to work. We do not try toconstruct the infinite-volume corrector as it would require to prove a quantitative homogenization theoremand establish a large-scale regularity theory (following the techniques of [ , Chapter 3]) and the developmentof this technology is unnecessary to prove Theorem 1. Nevertheless, the specific structure of the problem (andthe strong regularity properties established in Chapter 5) allows to define the gradient of the infinite-volumecorrector; the construction is carried out in Proposition 4.4. This section is devoted to the definition and the study of the finite-volume corrector.
Definition . For each integer m ∈ N , and each slope p ∈ R d ×( d ) , we definethe finite-volume corrector at scale 3 n to be the function χ n,p ∶ Z d × Ω → R ( d ) defined by the formula χ n,p ∶= u (⋅ , ⋅ , ◻ − n , p ) − l p . We recall that the corrector is equal to 0 outside the trimmed cube ◻ − n . Given two integers i ∈ { , . . . , d } and j ∈ { , . . . , ( d )} , we denote by e ij ∈ R d ×( d ) the vector e ij = ( , . . . , e i , . . . , ) , where the vector e i ∈ R d appears at the j -th position. We note that the collection of vectors ( e ij ) ≤ i ≤ d, ≤ j ≤( d ) is a basis of the vector space R d ×( d ) . We frequently refer to the corrector χ n,e ij by the notation χ n,ij . Remark . The finite volume corrector χ n,p is the solution of the equation ⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩ ∆ φ χ n,p − β ∆ χ n,p + β ∑ n ≥ β n (− ∆ ) n + χ n,p + ∑ q ∈Q ∇ ∗ q ⋅ a q ∇ q ( l p + χ n,p ) = ◻ − n ,χ n,p = ∂ ◻ − n . By the identity ∇ q ( l p + χ n,p ) , we see that the corrector depends only on the value of d ∗ l p . In particular, ifd ∗ l p = χ n,p = H − -norm of its flux. Proposition . There exist a constant C ∶= C ( d ) , an exponent α ( d ) > andan inverse temperature β ( d ) < ∞ such that for every inverse temperature β > β and every vector p ∈ R d ×( d ) ,the finite-volume corrector satisfies the following estimates (4.1) ∥ χ n,p ∥ L (◻ − n ,µ β ) ≤ C ∣ p ∣ ( − α ) n and (4.2) XXXXXXXXXXX ( p + ∇ χ n,p ) + β ∑ q ∈Q a q ∇ q ( l p + χ n,p ) L t , d ∗ ( n q ) − a p XXXXXXXXXXX H − (◻ − n ,µ β ) ≤ C ∣ p ∣ ( − α ) n . Proof.
The estimate (4.1) is obtained by combining Lemma 3.5 and Corollary 3.8. The proof of theestimate (4.2) regarding the flux is more involved and we split the argument into two steps. The argumentrequires to take into account the infinite range of the sum over the charges (by using the boundary layer BL n and the exponential decay of the coefficient a q ), which makes the proof technical. Since similar technicalitieshave already been treated in the previous sections and the analysis does not contain any new arguments, weomit some of the details and only write a (detailed) sketch of the proof. Step 1.
In this step, we prove that, to prove (4.2) is is sufficient to prove the estimate, for each p ∗ ∈ R d ×( d ) ,(4.3) XXXXXXXXXXX ∇ v (⋅ , ⋅ , ◻ n , p ∗ ) + β ∑ supp q ⊆◻ n a q ∇ q v (⋅ , ⋅ , ◻ n , p ∗ ) L t , d ∗ ( n q ) − p ∗ XXXXXXXXXXX H − (◻ − n ,µ β ) ≤ C ∣ p ∗ ∣ ( − α ) n . We fix a vector p ∗ ∈ R d ×( d ) and recall that, by definition of the first order corrector, l p + χ n,p = u (⋅ , ⋅ , ◻ − n , p ) . Toease the notation, we denote by u ∶= u (⋅ , ⋅ , ◻ − n , p ) and by v ∶= v (⋅ , ⋅ , ◻ n , a p ) . First, we note that Proposition 1.12 . DEFINITION OF THE FIRST-ORDER CORRECTOR AND QUANTITATIVE SUBLINEARITY 115 implies the inequality ∣ a (◻ − m ) − a ∣ ≤ C − αm . Combining this result with the estimate (1.7), we obtain theinequality, for each vector p ∈ R d ×( d ) ,(4.4) ∥∇ v (⋅ , ⋅ , ◻ n , a p ) − ∇ v (⋅ , ⋅ , ◻ n , a n p )∥ L (◻ n ,µ β ) = ∥∇ v (⋅ , ⋅ , ◻ n , a p − a n p )∥ L (◻ n ,µ β ) ≤ C − αn ∣ p ∣ . We use the inequality (4.4) with the estimate (3.50) stated in the proof of Proposition 3.6 and Corollary 3.8.We deduce that ∥∇ u − ∇ v ∥ L (◻ n ,µ β ) ≤ ∥∇ u − ∇ v (⋅ , ⋅ , ◻ n , a n p )∥ L (◻ n ,µ β ) + ∥∇ v (⋅ , ⋅ , ◻ n , a n p ) − ∇ v ∥ L (◻ n ,µ β ) (4.5) ≤ C − αn ∣ p ∣ . Using the estimate (4.5), we can write
XXXXXXXXXXX ∇ u + β ∑ q ∈Q a q (∇ q u ) L t , d ∗ ( n q ) − a p XXXXXXXXXXX H − (◻ − n ,µ β ) ≤ XXXXXXXXXXX ∇ v + β ∑ q ∈Q a q (∇ q v ) L t , d ∗ ( n q ) − a p XXXXXXXXXXX H − (◻ − n ,µ β ) + XXXXXXXXXXX ∇ ( u − v ) + β ∑ q ∈Q a q (∇ q ( u − v )) L t , d ∗ ( n q )XXXXXXXXXXX H − (◻ n ,µ β ) ≤ XXXXXXXXXXX ∇ v + β ∑ q ∈Q a q (∇ q v ) L t , d ∗ ( n q ) − p XXXXXXXXXXX H − (◻ n ,µ β ) + CR XXXXXXXXXXX ∇ ( u − v ) + β ∑ q ∈Q a q (∇ q ( u − v )) L t , d ∗ ( n q ) − p XXXXXXXXXXX L (◻ n ,µ β ) . Using the estimate (1.9) on the coefficient a q , we see that XXXXXXXXXXX ∇ ( u − v ) + β ∑ q ∈Q a q (∇ q ( u − v )) L t , d ∗ ( n q ) − p XXXXXXXXXXX L (◻ n ,µ β ) ≤ C ∥∇ ( u − v )∥ L (◻ n ,µ β ) ≤ C − αn ∣ p ∣ . A combination of the two previous displays shows(4.6)
XXXXXXXXXXX ∇ u + β ∑ q ∈Q a q (∇ q u ) n q − p XXXXXXXXXXX H − (◻ n ,µ β ) ≤ XXXXXXXXXXX ∇ v + β ∑ q ∈Q a q (∇ q v ) L t , d ∗ ( n q ) − a p XXXXXXXXXXX H − (◻ n ,µ β ) + C ( − α ) n ∣ p ∣ . The estimate (4.6) implies that to prove the inequality (4.2), it is sufficient to prove (4.3).
Step 2. Proving the estimate (4.3) . The argument is similar to the proof presented in Lemma 3.5. Toease the notation, we denote by v ∶= v (⋅ , ⋅ , ◻ n , p ∗ ) and by v z,m ∶= v (⋅ , ⋅ , z + ◻ m , p ∗ ) and assume without loss ofgenerality that ∣ p ∗ ∣ =
1. We use the H − -version of the multiscale Poincar´e inequality stated in Proposition 0.1of Appendix A. We obtain XXXXXXXXXXX ∇ v + β ∑ q ∈Q a q ∇ q vL t , d ∗ ( n q ) − p ∗ XXXXXXXXXXX H − (◻ − n ,µ β ) (4.7) ≤ C XXXXXXXXXXX ∇ v + β ∑ q ∈Q a q ∇ q vL t , d ∗ ( n q ) − p ∗ XXXXXXXXXXX L (◻ − n ,µ β ) + C n n ∑ m = ∑ z ∈Z m,n m ∣Z m,n ∣ ⟨⎛⎝ ∣ z + ◻ m ∣ ∑ x ∈ z +◻ m ∇ v ( x, ⋅) + β ∑ q ∈Q a q ∇ q vL t , d ∗ ( n q ( x )) − p ∗ ⎞⎠ ⟩ µ β .
16 6. QUANTITATIVE CONVERGENCE OF THE SUBADDITIVE QUANTITIES
The first term in the right side of (4.7) can be estimated by the estimate (1.7). We obtain(4.8)
XXXXXXXXXXX ∇ v + β ∑ q ∈Q a q ∇ q vL t , d ∗ ( n q ) − p ∗ XXXXXXXXXXX L (◻ − n ,µ β ) ≤ C. To estimate the second term in the right side of (4.7), we proceed as in Lemma 3.5 and use the subadditivityestimate stated in Proposition 2.5 and Corollary 2.6. We obtain ∑ z ∈Z m,n ∣Z m,n ∣ ⟨⎛⎝ ∣ z + ◻ m ∣ ∑ x ∈ z +◻ m ∇ v ( x, ⋅) + β ∑ q ∈Q a q ∇ q vL t , d ∗ ( n q ) − p ∗ ⎞⎠ ⟩ µ β (4.9) ≤ ∑ z ∈Z m,n ∣Z m,n ∣ ⟨⎛⎝ ∣ z + ◻ m ∣ ∑ x ∈ z +◻ m ∇ v z,m ( x, ⋅) + β ∑ q ∈Q a q ∇ q v z,m L t , d ∗ ( n q ( x )) − p ∗ ⎞⎠ ⟩ µ β + C − αm . We then use the two following results: ● One has the identity, for each point z ∈ Z m,n , ⟨ ∣ z + ◻ m ∣ ∑ x ∈ z +◻ m ⎛⎝ ∇ v z,m ( x, ⋅) + β ∑ q ∈Q a q ∇ q v z,m L t , d ∗ ( n q ( x ))⎞⎠⟩ µ β = p ∗ ; ● By Lemma 3.2, the inequality d − ≥ valid in dimension larger than 3 and the translation invarianceof the measure µ β , one has the variance estimatevar ⎡⎢⎢⎢⎢⎣ ∣ z + ◻ m ∣ ∑ x ∈ z +◻ m ⎛⎝ ∇ v z,m ( x, ⋅) + β ∑ q ∈Q a q ∇ q v z,m L t , d ∗ ( n q ( x ))⎞⎠⎤⎥⎥⎥⎥⎦ ≤ C − m . We obtain the estimate(4.10) ⟨⎛⎝ ∣ z + ◻ m ∣ ∑ x ∈ z +◻ m ∇ v z,m ( x, ⋅) + β ∑ q ∈Q a q ∇ q v z,m L t , d ∗ ( n q ( x )) − p ∗ ⎞⎠ ⟩ µ β ≤ C − m . Combining the estimates (4.7), (4.8), (4.9) and (4.10), we have obtained
XXXXXXXXXXX ∇ v + β ∑ q ∈Q a q ∇ q vL t , d ∗ ( n q ) − p ∗ XXXXXXXXXXX H − (◻ − n ,µ β ) ≤ C ( − α ) n . The proof of Proposition 4.3 is complete. (cid:3)
The next proposition establishes the existence andstationarity of the spatial gradient of the infinite-volume corrector.
Proposition . Thereexists a stationary random field ∇ χ ∶ Z d × Ω → R satisfying the following property, for each p ∈ R d and eachinteger n ∈ N , ∥∇ χ n,p − ∇ χ p ∥ L (◻ m ,µ β ) ≤ C − nα . Remark . The property stated in Remark 4.2 about the finite volume corrector also applies to theinfinite volume corrector: the function ∇ χ p depends only on the value of d ∗ l p . AS the vectors d ∗ l p belongto the space R d , the collection of correctors ( χ p ) p ∈ R d ×( d ) forms a d -dimensional vector space from which weextract a basis: for each integer i ∈ { , . . . , d } , we select a vector p ∈ R d ×( d ) such that d ∗ l p = e i and denote by ∇ χ i = ∇ χ p i .Let us first present the main idea of the argument. By assuming that the inverse temperature is largeenough, one has C , − ε -regularity estimates for the solutions of the Helffer-Sj¨ostrand equation, following thearguments given in Section 2 of Chapter 5. By Proposition 1.10, one also has an algebraic rate of convergencefor the subadditive energy ν with exponent α . The exponent ε depends on the inverse temperature β and tendsto 0 as β tends to infinity while the exponent α depends only on the dimension and remains unchanged bysending the inverse temperature to infinity. In other words, as the inverse temperature β tends to infinity, theregularity exponent ε tends to 0 and the exponent α remains bounded away from 0. It is thus possible to choose . DEFINITION OF THE FIRST-ORDER CORRECTOR AND QUANTITATIVE SUBLINEARITY 117 β sufficiently large so that the exponent ε is smaller than the exponent α and to leverage on this property, the C , − ε -regularity estimate presented in Proposition 2.2 of Chapter 5 and the Caccioppoli inequality to provethe existence of the gradient of the infinite-volume corrector. Proof.
We fix a vector p ∈ R d ×( d ) and assume without loss of generality that ∣ p ∣ =
1. We decompose theproof into two steps. In the first step, we prove that for each point x ∈ Z d , the sequence (∇ χ n,p ( x )) n ∈ N isCauchy in the space L ( µ β ) . This implies that it converges and we define the gradient of the infinite-volumecorrector to be its limit. In the second step we prove that the function ∇ χ p is stationary. Step 1.
We prove the inequality, for each point x ∈ Z d integer n ∈ N such that x ∈ ◻ − n ,(4.11) ∥∇ χ n,p ( x, ⋅) − ∇ χ n + ,p ( x, ⋅)∥ L ( µ β ) ≤ C − α n . We now fix a point x ∈ Z d and prove the estimate (4.11). By the definition of the correctors stated inDefinition 4.1, and the definition of the function u as the minimizer in the definition of the energy quantity ν given in (1.3), we see that for each integer n ∈ N , the functions χ n and χ n + are solutions of the Helffer-Sj¨ostrandequations L ( l p + χ n,p ) = ◻ − n × Ω and
L ( l p + χ n + ,p ) = ◻ − n + × Ω . In particular, the difference χ n + ,p − χ n,p is solution of the equation L ( χ n + ,p − χ n,p ) = ◻ − m × Ω.We can thus apply Proposition 2.4 of Chapter 5 to obtain, for each integer n ∈ N such that x ∈ ◻ − n , ∥∇ χ n,p ( x, ⋅) − ∇ χ n + ,p ( x, ⋅)∥ L ( µ β ) ≤ sup y ∈◻ − n ∥∇ χ n,p ( y, ⋅) − ∇ χ n + ,p ( y, ⋅)∥ L ( µ β ) (4.12) ≤ C ( ε − ) n ∥ χ n,p − χ n + ,p − ( χ n,p − χ n + ,p ) ◻ − n ∥ L (◻ − n ,µ β ) ≤ C ( ε − ) n ∥ χ n,p − χ n + ,p ∥ L (◻ − n ,µ β ) . By combining the estimate (4.12) and Proposition 4.3, we obtain the estimate, for each pair of integers n ∈ N such that x ∈ ◻ − n , ∥∇ χ n,p ( x, ⋅) − ∇ χ n + ,p ( x, ⋅)∥ L ( µ β ) ≤ C ( ε − α ) n . Using the assumption ε ≤ α , we obtain(4.13) ∥∇ χ n,p ( x, ⋅) − ∇ χ n + ,p ( x, ⋅)∥ L ( µ β ) ≤ C − α n . The inequality (4.13) implies that, the sequence (∇ χ n,p ) n ∈ N is Cauchy in the space L ( µ β ) . This implies thatit converges in the space L ( µ β ) . We define the gradient of the corrector ∇ χ p ( x ) to be the limiting object.From the estimate (4.13), we also deduce that it satisfies the inequality, for each pair of integers n ∈ N , ∥∇ χ n,p ( x, ⋅) − ∇ χ p ( x, ⋅)∥ L ( µ β ) ≤ C − α n . The proof of Step 1 is complete.
Step 2.
In this step, we prove the stationarity of the infinite-volume gradient corrector. For z ∈ Z d , werecall the notation τ z for the translation of the field introduced in Chapter 2. We prove the identity, for each ( x, φ ) ∈ Z d × Ω,(4.14) ∇ χ p ( x, φ ) = ∇ χ p ( z + x, τ z φ ) . To prove the equality (4.14), we first note that, by the definition of the function u , we have the equality, foreach point z ∈ Z d , each cube ◻ ⊆ Z d , and each pair ( x, φ ) ∈ ( y + ◻) × Ω,(4.15) u ( x, φ, y + ◻ , p ) = u ( x − y, τ − y φ, ◻ , p ) . Using the identity (4.15), the result established in Step 1 and the translation invariance of the measure µ β , we obtain that the sequence (∇ u ( x, ⋅ , y + ◻ n , p ) − p ) n ∈ N converges in the space L ( µ β ) to the randomvariable φ → ∇ χ p ( x − y, τ − y φ ) . Thus to prove the identity (4.14), it is sufficient to prove that the sequence (∇ u ( x, ⋅ , y + ◻ n , p ) − p ) n ∈ N also converges in L ( µ β ) to the gradient of the corrector φ → ∇ χ p ( x, φ ) . This iswhat we now prove.We first note that the proof Proposition 1.12 can be adapted so as to have the following result. For each y ∈ Z d and each integer n such that 3 n ≥ ∣ y ∣ , one has the estimate(4.16) ∑ z ∈Z n ∥∇ u (⋅ , ⋅ , y + z + ◻ n , p ) − ∇ u (⋅ , ⋅ , ◻ n + , p )∥ L ( y + z +◻ n ,µ β ) ≤ C ( ν (◻ n , p ) − ν (◻ n + , p ) + − n ) .
18 6. QUANTITATIVE CONVERGENCE OF THE SUBADDITIVE QUANTITIES
The proof is identical; indeed under the assumption 3 n ≥ ∣ y ∣ , one can partition the triadic cube ( y + ◻ n + ) into the collection of triadic cubes ( y + z + ◻ n ) z ∈Z n and a boundary layer of width of size 3 n . One can thenrewrite the proof of Proposition 1.12 to obtain the estimate (4.16). We then use Proposition 1.12 (or moreprecisely Corollary 3.8) and obtain the inequality ∥∇ u (⋅ , ⋅ , y + ◻ n , p ) − ∇ u (⋅ , ⋅ , ◻ n + , p )∥ L ( y +◻ n ,µ β ) ≤ C − αn . Using the C − ε -regularity estimate stated in Proposition 2.4, the assumption ε ≤ α and an argument similar tothe one presented in Step 1, we obtain, for each integer n ∈ N such that 3 n ≥ ∣ y ∣ and each point x ∈ ◻ − n , ∥∇ u ( x, ⋅ , y + ◻ n , p ) − ∇ u ( x, ⋅ , ◻ n + , p )∥ L ( µ β ) ≤ C − α n . Using the definition of the finite-volume corrector given in Definition 4.1 and the inequality (4.11), we deducethat ∥∇ u ( x, ⋅ , y + ◻ n , p ) − p − ∇ χ p ( x, ⋅)∥ L ( µ β ) ≤ C − α n . The previous inequality implies that the sequence (∇ u ( x, ⋅ , y + ◻ n , p ) − p ) n ∈ N converges in the space L ( µ β ) to the random variable φ → ∇ χ p ( x, φ ) . The proof of Proposition 4.4 is complete. (cid:3) HAPTER 7
Quantitative homogenization of the Green’s matrix
1. Statement of the main result
The objective of this chapter is to prove the homogenization of the mixed gradient of the Green’s matrix.We first introduce the notation a β ∶= a β and the Green’s matrix associated to the homogenized operator ∇ ⋅ a β ∇ :we denote by G ∶ Z d → R ( d )×( d ) the fundamental solution of the elliptic system(1.1) − ∇ ⋅ a β ∇ G = δ in Z d . The matrix a β is a small perturbation of the matrix β I d and the size of the perturbation is of order β − ≪ β − .The solvability of the equation is thus ensured by the arguments of Chapter 5; more specifically, a Nash-Aronsonestimate holds for the heat-kernel associated to the operator −∇ ⋅ a β ∇ (which implies the solvability by anintegration in time of the heat in dimension larger than 3).The following theorem is the main result of this chapter. Theorem . Fix a charge q ∈ Q suchthat belongs to the support of n q and let U q be the solution of the Helffer-Sj¨ostrand equation (1.2) LU q = cos 2 π ( φ, q ) q in Z d × Ω . for each integer k ∈ { , . . . , ( d )} , we define the function G q ,k ∶ Z d → R by the formula G q ,k = ∑ ≤ i ≤ d ∑ ≤ j ≤( d ) ⟨ cos 2 π ( φ, q ) ( n q , d ∗ l e ij + d ∗ χ ij )⟩ µ β ∇ i G jk . Then there exist an inverse temperature β ∶= β ( d ) < ∞ , an exponent γ ∶= γ ( d ) > and a constant C q whichsatisfies the estimate ∣ C q ∣ ≤ C ∥ q ∥ k for some C ∶= C ( d, β ) < ∞ and k ∶= k ( d ) < ∞ , such that for each β ≥ β and each radius R ≥ , one has the inequality (1.3) XXXXXXXXXXXXX∇U q − ∑ ≤ i ≤ d ∑ ≤ j ≤( d ) ( e ij + ∇ χ ij ) ∇ i G q ,j XXXXXXXXXXXXX L ( A R ,µ β ) ≤ C q R d + γ . Remark . Since the codifferential d ∗ is a linear functional of the gradient, the map d ∗ χ ij is well-defined even if we have only defined the gradient of the infinite-volume corrector: we have the formulad ∗ χ ij ∶= L , d ∗ (∇ χ ij ) . Remark . We recall that in this chapter, the constants are allowed to depend on the dimension d andon the inverse temperature β . Remark . We recall the definition of the annulus A R ∶= B R ∖ B R . The volume of the annulus A R isof order R d . Remark . The double sum ∑ ≤ i ≤ d ∑ ≤ j ≤( d ) appears frequently in the proofs of this chapter; to ease thenotation, we denote it by ∑ i,j . Remark . Since the the form q can be written d n q , we expect the two gradients ∇U q and ∇ G q to behave like mixed derivatives of the Green’s matrix, i.e., they should be of order R − d in the annulus A R . The proposition asserts that the difference between the two terms ∇U q and ∑ i,j ( e ij + ∇ χ ij ) ∇ i G q ,j isquantitatively smaller than the typical size of the two terms considered separately.
2. Outline of the argument
The strategy of the proof of Theorem 4 relies on a classical strategy in homogenization: the two-scale-expansion. The proofs presented in the chapter make essentially use of two ingredients established in Chapters 5and 6: ● The quantitative sublinearity of the finite-volume corrector and the estimate on the H − -norm of theflux stated in Proposition 4.3 of Chapter 6; ● The C , − ε -regularity theory established in Chapter 5.We now give an outline of the proof of Theorem 4. The argument is split into two sections: ● In Section 3, we perform the two-scale expansion and obtain a result of homogenization for thegradient of the Green’s matrix as stated in Proposition 2.1; ● In Section 4, we use the result of Proposition 2.1 and perform the two-scale expansion a second timeto obtain the quantitative homogenization of the mixed derivative of the Green’s matrix stated inTheorem 4.
In Section 3, we prove the homoge-nization of the gradient of the Green’s matrix stated in Proposition 2.1 below.
Proposition . Let
G ∶ Z d × Ω → R ( d )×( d ) be the Green’s matrixassociated to the Helffer-Sj¨ostrand equation (2.1) LG = δ in Z d × Ω . Then there exist an inverse temperature β ( d ) < ∞ , an exponent γ ∶= γ ( d ) > and a constant C ∶= C ( d ) < ∞ such that for any β > β , any radius R ≥ , for any of integer k ∈ { , . . . , ( d )} , (2.2) XXXXXXXXXXX∇G ⋅ k − ∑ i,j ( e ij + ∇ χ ij ) ∇ i G jk XXXXXXXXXXX L ( A R ,µ β ) ≤ CR d − + γ . The argument relies on a two-scale expansion. To set up the argument, we first select an inverse temperature β large enough, depending only on the dimension d , such that the quantitative sublinearity of the finite-volumecorrector and of its flux stated in Proposition 4.3 of Chapter 6 hold with exponent α >
0. Following theargument explained at the beginning of Section 4 there, we can choose the parameter β large enough so that allthe results presented in Chapter 5 pertaining to the C , − ε -regularity theory for the Helffer-Sj¨ostrand operator L are valid with a regularity exponent ε which is small compared to the exponent α (we assume for instancethat the ratio between ε and α is smaller than 100 d ). We also fix an exponent δ which is both larger than ε and smaller than α and corresponds to the size of a mollifier exponent which needs to be taken into accountin the argument (we assume for instance that the ratios between the exponents α and δ and between theexponents ε and δ are both smaller than 10 d ). To summarize, we have three exponents in the argument, whichcan be ordered by the following relations(2.3) 0 < ε ·‚¶ regularity ≪ δ ·‚¶ mollifier exponent ≪ α ·‚¶ homogenization ≪ . We additionally assume that the exponents ε , δ and α are chosen in a way that they depend only on thedimension d . The exponent γ in the statement of Proposition 3.1 depends only ε , δ and α (and thus only onthe dimension d ).We now give an outline of the proof of the inequality (2.2); the details can be found in Step 1 of the proofof Proposition 3.1. The first step of the argument is to approximate the Green’s matrices G and G ; the mainissue is that the spatial Dirac function δ in the definitions of the Green’s matrices G in (2.1) and G in (1.1) istoo singular and causes some problems in the analysis. To remedy this issue, we replace the Dirac function δ by a smoother function and make use of the boundary layer exponent δ : we let ρ δ be a discrete function from Z d to R ( d )×( d ) , we denote its components by ( ρ δ,ij ) ≤ i,j ≤( d ) and assume that they satisfy the four properties(2.4) supp ρ δ ⊆ B R − δ , ≤ ρ δ,ij ≤ CR −( − δ ) d , ∑ x ∈ Z d ρ δ,ij ( x ) = { i = j } , and ∀ k ∈ N , ∣∇ k ρ δ,ij ∣ ≤ CR ( d + k )( − δ ) , . OUTLINE OF THE ARGUMENT 121 and define the functions G δ ∶ Z d × Ω → R ( d )×( d ) and G δ ∶ Z d → R ( d )×( d ) to be the solution of the systems, foreach integer k ∈ { , . . . , ( d )} (2.5) LG δ, ⋅ k = ρ δ, ⋅ k in Ω × Z d , −∇ ⋅ ( a β ∇ G δ, ⋅ k ) = ρ δ, ⋅ k in Z d , We then prove, by using the C , − ε -regularity theory established in Section 4 of Chapter 5, that the functions G δ , G δ are good approximations of the functions G , G . This is the subject of Lemma 3.1 where we prove thatfor β sufficiently large, there exists an exponent γ ∶= γ ( d, β, δ, ε ) > ∥∇G δ − ∇G∥ L ∞ ( A R ,µ β ) ≤ CR − d − γ and ∥∇ G δ − ∇ G ∥ L ∞ ( A R ,µ β ) ≤ CR − d − γ . By the estimates (2.6), we see that to prove Proposition 3.1 it is sufficient to prove the inequality, for eachinteger k ∈ { , . . . , ( d )} ,(2.7) XXXXXXXXXXX∇G δ, ⋅ k − ∑ i,j ( e ij + ∇ χ ij ) ∇ i G δ,jk XXXXXXXXXXX L ( A R ,µ β ) ≤ CR − d − γ . We now sketch the proof of the inequality (2.7). We let m be the integer uniquely defined by the inequalities3 m ≤ R + δ < m + , consider the collection finite-volume correctors ( χ m,ij ) ≤ i ≤ d, ≤ j ≤( d ) . We then define thetwo-scale expansion H δ ∶ Z d × Ω → R ( d )×( d ) according to the formula, for each integer k ∈ { , . . . , ( d )} ,(2.8) H δ, ⋅ k ∶= G δ, ⋅ k + ∑ i,j (∇ i G δ,jk ) χ m,ij . We fix an integer k ∈ { , . . . , ( d )} . The strategy is to compute the value of LH δ, ⋅ k by using the explicit formulaon H δ, ⋅ k stated in (2.8) and to prove that it is quantitatively close to the map ρ δ, ⋅ k in the correct functionalspace; precisely, we prove the H − -estimate, for each integer k ∈ { , . . . , ( d )} (2.9) ∥LH δ, ⋅ k − ρ δ, ⋅ k ∥ H − ( B R + δ,µβ ) ≤ CR − d − γ . Obtaining this result relies on the quantitative behavior of the corrector and of the flux established inProposition 4.3 of Chapter 6. Once one has a good control over the H − -norm of LH δ, ⋅ k − ρ δ, ⋅ k , the inequality (2.7)can be deduced from the following two arguments: ● We use that the function G δ, ⋅ k satisfies the equation LG δ, ⋅ k = ρ δ, ⋅ k to obtain that the H − -norm of theterm L (H δ, ⋅ k − G δ, ⋅ k ) is small. We then introduce a cutoff function η ∶ Z d → R which satisfies:supp η ⊆ A R , ≤ η ≤ , η = { x ∈ Z d ∶ . R ≤ ∣ x ∣ ≤ . R } , and ∀ k ∈ N , ∣∇ k η ∣ ≤ CR k , and use the function η (H δ, ⋅ k − G δ, ⋅ k ) as a test function in the definition of the H − -norm of theinequality (2.9). We obtain that the L -norm of the difference (∇H δ, ⋅ k − ∇G δ, ⋅ k ) is small (the cutofffunction is used to ensure that the function η (H δ, ⋅ k − G δ, ⋅ k ) is equal to 0 on the boundary of the ball B R + δ and can thus be used as a test function). The precise estimate we obtain is written below(2.10) ∥∇H δ, ⋅ k − ∇G δ, ⋅ k ∥ L ( Z d ,µ β ) ≤ CR d + − d − γ ; ● By using the formula (2.8), we can compute an explicit formula for the gradient of the two-scaleexpansion H δ, ⋅ k . We then use the quantitative sublinearity of the corrector stated in Proposition 4.3,Chapter 6 and the property of the gradient of the infinite volume corrector stated in Proposition 4.4to deduce that, for each integer k ∈ { , . . . , ( d )} , the L -norm of the difference ∇H δ, ⋅ k − ∑ i,j ( e ij +∇ χ ij )∇ i G jk is small; the precise result we obtain is the following(2.11) XXXXXXXXXXX∇H δ, ⋅ k − ∑ i,j ( e ij + ∇ χ ij )∇ i G jk XXXXXXXXXXX L ( Z d µ β ) ≤ CR d + − d − γ . The inequality (2.7) is then a consequence of the inequalities (2.10) and (2.11).
22 7. QUANTITATIVE HOMOGENIZATION OF THE GREEN’S MATRIX
In Section 4, we useProposition 2.1 to prove Theorem 4. The argument is decomposed into three steps: ● In Step 1, we use Proposition 2.1 and the symmetry of the Helffer-Sj¨ostrand operator L to prove theinequality in expectation(2.12) ⎛⎝ R − d ∑ z ∈ A R ∣⟨U q ( z, ⋅)⟩ µ β − G q ( z )∣ ⎞⎠ ≤ CR d − + γ . ● In Step 2, we prove the variance estimate, for each point z ∈ Z d ,(2.13) var [U q ( z, ⋅)] ≤ C q ∣ z ∣ d − ε . Since we expect the function z ↦ U q ( z ) to decay like ∣ z ∣ − d ; its variance should be of order ∣ z ∣ − d .The estimate (2.13) states that the variance of the random variable φ → U q ( z, φ ) is (quantitatively)smaller than its size; this means that the random variable U q ( z ) concentrates on its expectation.We then use the result established in Step 1 to refine the result: since by (2.12), one knows that theexpectation of U q ( z ) is close to the function G q , one deduces that the function U q is close to thefunction G q in the L ( A R , µ β ) -norm. The precise estimate we obtain is the following(2.14) ∥U q − G q ∥ L ( A R ,µ β ) ≤ C q R d − − γ . The proof of the inequality (2.13) does not rely on tools from stochastic homogenization; we appealto the Brascamp-Lieb inequality and used the differentiated Helffer-Sj¨ostrand introduced in Section 4of Chapter 5. ● In Step 3, we prove the estimate (1.3), the proof is similar to the argument presented in the proof ofProposition 2.1 and relies on a two-scale expansion. The argument is split into three substeps. Wefirst define the two-scale expansion H q by the formula(2.15) H q ∶= G q + ∑ i,j ∇ i G q ,j χ m,ij . We can then use that the function G q is the solution to the homogenized equation a ∆ G q = A R to prove that the H − ( A R , µ β ) -norm of the term LH q over the annulus A R is small;this is the subject of Substep 3.1 where we prove(2.16) ∥LH q ∥ H − ( A R ,µ β ) ≤ C q R d + γ . The proof is essentially a notational modification of the proof of the estimate (2.9) (and is evensimpler since we do not have to take into account the exponent δ and the function ρ δ ). Once wehave proved that the inequality (2.16), we use that the function U q satisfies the identity LU q = A R × Ω to deduce that the H − ( A R , µ β ) -norm of the term L (H q − U q ) = LH q is small. Wethen introduce the annulus A R ∶= { x ∈ Z d ∶ . R ≤ ∣ x ∣ ≤ . R } and a cutoff function η satisfying theproperties: supp η ⊆ A R , ≤ ρ ≤ , η = A R , and ∀ k ∈ N , ∣∇ k η ∣ ≤ CR k . We note that the choice of the values 1 . . A R is arbitrary; anypair of real numbers belonging to the interval ( , ) would be sufficient for our purposes. We thenuse the function η (H q − U q ) as a test function in the definition of the H − ( A R , µ β ) -norm of theterm L (H q − U q ) and use the L ( A R , µ β ) -estimate (2.14) to obtain that the L ( A R , µ β ) -norm ofthe difference ∇H q − ∇U q is small. This is the subject of Substep 3.2 where we prove(2.17) ∥∇U q − ∇H q ∥ L ( A R ,µ β ) ≤ C q R d + γ . ● Step 4 is the conclusion of the argument, we use the explicit formula for the two-scale expansion H q given in (2.15), the quantitative sublinearity of the corrector stated in Proposition 4.3 of Chapter 6 . TWO-SCALE EXPANSION AND HOMOGENIZATION OF THE GRADIENT OF THE GREEN’S MATRIX 123 and the quantitative estimate for the difference of the finite and infinite-volume gradient of thecorrector stated in Proposition 4.4 there to prove the estimate(2.18) XXXXXXXXXXX∇H q − ∑ i,j ( e ij + ∇ χ ij ) ∇ i G q ,j XXXXXXXXXXX L ( A R ,µ β ) ≤ C q R d + γ . The argument is a notational modification of the one used to prove (2.15). We finally combine theestimates (2.17) and (2.18) to obtain the estimate (1.3) and complete the proof of Theorem 4.
3. Two-scale expansion and homogenization of the gradient of the Green’s matrix
This section is devoted to the proof of Proposition 2.1. We collect some preliminary results in Section 3.1and prove Theorem 4 in Sections 3.2, 3.3 and 3.4 following the outline given in Section 2.2.
In this section, we collect some preliminary properties which are used inthe proof of Proposition 2.1.3.1.1.
Notations for the exponent γ . We first introduce some notations for the exponent γ . As was alreadymentioned, this exponent depends on the parameters α, δ and ε ; in the argument, we need to keep track of itstypical size and proceed as follows: ● We use the notation γ when the exponent is of order 1; a typical example is the exponent γ α ∶= − c α − c δ − c ε for some constants c , c , c depending only on the dimension d ; ● We use the notation γ α when the exponent is of order α ; a typical example is the exponent γ α ∶= α − c δ − c ε for some constants c , c depending only on the dimension d ; ● We use the notation γ δ when the exponent is of order δ ; a typical example is the exponent γ δ ∶= δ − c ε for some constant c depending only on the dimension d .We always have the ordering 0 < γ ε ≪ γ δ ≪ γ α ≪ γ . We also allow the value of the exponents γ ε , γ δ , γ α , γ to vary from line to line in the argument as long as theorder of magnitude is preserved. In particular, we may write γ = γ − α, γ α = γ α − δ and γ δ = γ δ − ε. Regularity estimates.
In this section, we record some regularity estimates pertaining to the Green’smatrices G , G δ , G and G δ . Proposition . The following properties hold: ● There exists an exponent γ δ > such that one has the estimates (3.1) ∥∇G( x, ⋅) − ∇G δ ( x, ⋅)∥ L ∞ ( A R ,µ β ) ≤ CR d − + γ δ and ∥∇ G − ∇ G δ ∥ L ∞ ( A R ) ≤ CR d − + γ δ ; ● The Green’s matrix G δ satisfies the following L ∞ -estimates (3.2) ∥G δ ∥ L ∞ ( Z d ,µ β ) ≤ CR ( − δ )( d − ) and ∥∇G δ ∥ L ∞ ( Z d ,µ β ) ≤ CR ( − δ )( d − − ε ) , as well as the estimates (3.3) ∥G δ ∥ L ∞ ( A R + δ ,µ β ) ≤ CR ( + δ )( d − ) and ∥∇G δ ∥ L ∞ ( A R + δ ,µ β ) ≤ CR ( + δ )( d − − ε ) ; ● The homogenized Green’s matrix G δ satisfies the regularity estimate, for each integer k ∈ N , (3.4) ∥∇ k G δ ∥ L ∞ ( Z d ,µ β ) ≤ CR ( − δ )( d − + k ) , as well as the estimate (3.5) ∥∇ k G δ ∥ L ∞ ( A R + δ ,µ β ) ≤ CR ( + δ )( d − + k ) .
24 7. QUANTITATIVE HOMOGENIZATION OF THE GREEN’S MATRIX
Proof of Proposition 3.1.
The proof relies on the regularity estimates established in Chapter 5. Wefirst note that, by definitions of the functions G and G δ , we have the identities(3.6) G ( x, φ ) = G ( x, φ ; 0 ) and G δ ( x, φ ) = ∑ y ∈ B R − δ G ( x, φ ; y ) ρ δ ( y ) , where the product in the right side of (3.6) is the standard matrix product between G ( x, φ ; y ) and ρ δ ( y ) . Usingthat the map ρ δ has total mass 1 and the regularity estimate on the Green’s matrix stated in Proposition 4.2of Chapter 5, we obtain, for each point x in the annulus A R , ∥∇ x G( x, φ ; 0 ) − ∇ x G δ ( x, φ ; y )∥ L ∞ ( µ β ) ≤ ∑ y ∈ B R − δ ρ δ ( y ) ∥∇ x G ( x, φ ; 0 ) − ∇ x G ( x, φ ; y )∥ L ∞ ( µ β ) ≤ R − δ sup y ∈ B R − δ ∥∇ x ∇ y G( x, φ ; y )∥ L ∞ ( µ β ) ≤ R − δ sup y ∈ B R − δ ∣ x − y ∣ − d − ε ≤ R − δ R − d − ε . This computation implies the estimate (3.1) with the exponent γ δ = δ − ε which is strictly positive by theassumption (2.3).The estimate on the homogenized Green’s matrix is similar and even simpler since we only have to workwith the Green’s matrix associated to the discrete Laplacian on Z d (and not the Helffer-Sj¨ostrand operator L );we omit the details.The proof of the inequality (3.2) relies on the estimates on the Green’s matrix and its gradient establishedin Corollary 3.8 of Chapter 5. We use the identity (3.6) and write, for each point x ∈ Z d , ∥G δ ( x, ⋅)∥ L ∞ ( µ β ) = ∑ y ∈ B R − δ ∣ ρ δ ( y )∣ ∥G ( x, φ ; y )∥ L ∞ ( µ β ) ≤ R ( − δ ) d ∑ y ∈ B R − δ C ∣ x − y ∣ d − ≤ R ( − δ ) d ∑ y ∈ B R − δ C ∣ y ∣ d − ≤ R ( − δ )( d − ) . A similar computation shows the bound for the gradient of the Green’s matrix and the bounds (3.3) in theannulus A R + δ .To prove the regularity estimate (3.4), we use the definition of the map G δ given in (2.5) and note that −∇ ⋅ a β ∇ (∇ k G δ ) = ∇ k ρ δ in Z d . We then use the the properties of the function ρ δ stated in (2.4) and the standard estimates on the homogenizedGreen’s matrix G . We obtain, for each point x ∈ Z d , ∣∇ k G δ ( x )∣ ≤ ∑ y ∈ B R − δ ∣∇ k ρ δ ( y )∣ ∣ G ( x − y )∣ ≤ CR ( d + k )( − δ ) ∑ y ∈ B R − δ ∣ x − y ∣ d − ≤ CR ( − δ )( d − + k ) . There only remains to prove the estimate (3.5). We select a point x ∈ A R + δ and write ∣∇ k G δ ( x )∣ = RRRRRRRRRRRR ∑ y ∈ B R − δ ∇ k G ( x − y ) ρ δ ( y )RRRRRRRRRRRR ≤ ∑ y ∈ B R − δ ∣ ρ δ ( y )∣∣ x − y ∣ d − + k ≤ CR ( d − + k )( + δ ) ∑ y ∈ B R − δ ∣ ρ δ ( y )∣≤ CR ( + δ )( d − + k ) . (cid:3) We have now collected all the necessary preliminary ingredients for the proof of Proposition 2.1 and devotethe rest of Section 3 to its demonstration. LH δ − ρ δ . In this section, we fix an integer k ∈ { , . . . , ( d )} , let H δ, ⋅ k be the two-scale expansion introduced in (2.8) and prove that there exists an exponent γ α > ∥LH δ, ⋅ k − ρ δ, ⋅ k ∥ H − ( B R + δ ,µ β ) ≤ CR d − + γ α . . TWO-SCALE EXPANSION AND HOMOGENIZATION OF THE GRADIENT OF THE GREEN’S MATRIX 125 The strategy is to use the explicit formula for the map H δ, ⋅ k to compute the value of the term LH δ, ⋅ k . Wethen prove that its H − ( B R + δ , µ β ) -norm is small by using the quantitative properties of the corrector statedin Proposition 4.3 of Chapter 6. We first write(3.8) LH δ, ⋅ k = ∆ φ H δ, ⋅ k ·„„„„„„„„„„„„„‚„„„„„„„„„„„„¶ Substep 1.1 + β ∑ n ≥ β n (− ∆ ) n + H δ, ⋅ k ·„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„¶ Substep 1.2 − β ∆ H δ, ⋅ k + ∑ q ∈Q ∇ ∗ q ⋅ a q ∇ q H δ, ⋅ k ·„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„¶ Substep 1.3 . We treat the three terms in the right side in three distinct substeps.
Substep 1.1
In this substep, we treat the term ∆ φ H δ, ⋅ k . Since the homogenized Green’s matrix G δ, ⋅ k doesnot depend on the field φ , we have the formula(3.9) ∆ φ H δ, ⋅ k = ∑ i,j ∇ i G δ,jk ( ∆ φ χ m,ij ) . Sustep 1.2.
In this substep, we study the iteration of the Laplacian of the two-scale expansion. We provethe identity(3.10) ∑ n ≥ β n (− ∆ ) n + H δ, ⋅ k = ∑ i,j ∑ n ≥ β n ∇ i G δ,jk (− ∆ ) n + χ m,ij + R ∆ n , where R ∆ n is an error term which satisfies the H − ( B R + δ , µ β ) -estimate(3.11) ∥ R ∆ n ∥ H − ( B R + δ ,µ β ) ≤ CR d − + γ α . We use the following identity for the iteration of the Laplacian on a product of functions: given two smoothfunctions f, g ∈ C ∞ ( R d ) , we have the identity(3.12) ∆ n ( f g ) = n ∑ r = r ∑ l = ( n − rl ) (∇ r ∆ l f ) ⋅ (∇ r ∆ n − r − l g ) . We note that this formula is valid for continuous functions (with the continuous Laplacian), it can be adaptedto the discrete setting by taking into considerations translations of the functions f and g . Since this adaptationdoes not affect the overall strategy of the proof, we ignore this technical difficulty in the rest of the argumentand apply the formula (3.12) to the two-scale expansion H δ, ⋅ k as such. We obtain(3.13) ∆ n H δ, ⋅ k = ∆ n G δ, ⋅ k + ∑ i,j n ∑ r = r ∑ l = ( n − rl ) (∇ r ∆ l ∇ i G δ,jk ) ⋅ (∇ r ∆ n − r − l χ m,ij ) . We first focus on the term ∆ n G δ, ⋅ k in the identity (3.13) and prove that it is small in the H − ( B R + δ , µ β ) -norm.Using the regularity estimate (3.4), we have, for each integer n ≥ ∥ ∆ n G δ, ⋅ k ∥ H − ( B R + δ ,µ β ) ≤ CR + δ ∥ ∆ n G δ, ⋅ k ∥ L ( B R + δ ,µ β ) ≤ CR + δ ∥ ∆ n G δ, ⋅ k ∥ L ∞ ( B R + δ ) (3.14) ≤ C n R + δ R ( − δ )( d − + n ) ≤ C n R + δ R ( − δ )( d − + ) ≤ C n R d − + γ , where we have set γ ∶= + δ ( d + ) > G δ . We obtain the followinginequality: for each pair of integers ( i, j ) ∈ { , . . . , d } × { , . . . , ( d )} and each pair of integers ( r, l ) ∈ { , . . . , d }
226 7. QUANTITATIVE HOMOGENIZATION OF THE GREEN’S MATRIX such that l ≤ n − k and k + l ≥ ∥(∇ r ∆ l ∇ i G δ,jk ) ⋅ (∇ r ∆ n − r − l χ m,ij )∥ H − ( B R + δ ,µ β ) (3.15) ≤ CR + δ ∥(∇ r ∆ l ∇ i G δ,jk ) ⋅ (∇ r ∆ n − r − l χ m,ij )∥ L ( B R + δ ,µ β ) ≤ CR + δ ∥∇ r ∆ l ∇ i G δ,jk ∥ L ∞ ( B R + δ ) × ∥∇ r ∆ n − r − l χ m,ij ∥ L ( B R + δ ,µ β ) ≤ C r + l R + δ R ( − δ )( d − + l + r ) ∥∇ r ∆ n − r − l χ m,ij ∥ L ( B R + δ ,µ β ) . We use that the discrete operator ∇ r ∆ n − r − l is bounded in the space L ( B R + δ ) and Proposition 4.3 of Chapter 6to estimate the L -norm of the corrector. We obtain(3.16) ∥∇ r ∆ n − r − l χ m,ij ∥ L ( B R + δ ,µ β ) ≤ C n − l ∥ χ m,ij ∥ L ( B R + δ ,µ β ) ≤ C n − l R ( + δ )( − α ) . Putting the estimate (3.15) and (3.16) together and using the inequality 3 ≤ l + r ≤ n , we deduce that(3.17) ∥(∇ r ∆ l ∇ i G δ,jk ) ⋅ (∇ r ∆ n − r − l χ m,ij )∥ H − ( B R + δ ,µ β ) ≤ C n R + δ R ( − δ )( d + ) R ( + δ )( − α ) ≤ C n R d − + γ , where we have set γ ∶= + α − αδ + δ ( d − ) + δ > H − -norm of the terms corresponding to the parameters r = l = h ∈ H ( B R + δ , µ β ) such that ∥ h ∥ H ( B R + δ ,µ β ) ≤
1. We use the function h as a test function, perform an integration by parts in the first line,use the Cauchy-Schwarz inequality in the second line and the continuity of the discrete Laplacian in the thirdline. We obtain 1 R ( + δ ) d ∑ x ∈ B R + δ ⟨(∇∇ i G δ,jk ( x, ⋅)) ⋅ (∇ ∆ n − χ m,ij ( x, ⋅)) h ( x, ⋅)⟩ µ β (3.18) = R ( + δ ) d ∑ x ∈ B R + δ ⟨ χ m,ij ( x, ⋅)∇ ⋅ ∆ n − ((∇∇ i G δ,jk ( x, ⋅)) h ( x, ⋅))⟩ µ β ≤ ∥ χ m,ij ∥ L ( B R + δ ,µ β ) ∥∇ ⋅ ∆ n − r (∇∇ i G δ,jk h )∥ L ( B R + δ ,µ β ) ≤ C n ∥ χ m,ij ∥ L ( B R + δ ,µ β ) ∥∇ ⋅ (∇∇ i G δ,jk h )∥ L ( B R + δ ,µ β ) . Using the regularity estimate for the homogenized Green’s matrix stated in (3.4) and the inequality ∥ h ∥ L ( B R + δ ,µ β ) ≤ CR + δ (which is a consequence of the assumption ∥ h ∥ H ( B R + δ ,µ β ) ≤ ∥∇ ⋅ ((∇∇ i G δ,jk ) h )∥ L ( B R + δ ,µ β ) ≤ ∥∇ G δ,jk h ∥ L ( B R + δ ,µ β ) + ∥∇ G δ,jk ∇ h ∥ L ( B R + δ ,µ β ) (3.19) ≤ CR + δ R ( − δ )( d + ) + CR ( − δ ) d ≤ CR d − δ ( d + ) . We then combine the estimate (3.18) with the inequality (3.19) and the quantitative sublinearity of thecorrector stated in Proposition 4.3 of Chapter 6. We obtain(3.20) 1 R ( + δ ) d ∑ x ∈ B R + δ ⟨(∇∇ i G δ,jk ( x )) ⋅ (∇ ∆ n − χ m,ij ( x, ⋅)) h ( x, ⋅)⟩ µ β ≤ C n R ( + δ )( − α ) R d − δ ( d + ) ≤ C n R d − + γ α , where we have set γ α ∶= α ( + δ ) − δ ( d + ) > β large enough so that the series ( C n β n ) n ∈ N is summable, we obtain the main result (3.10)and (3.11) of this substep. . TWO-SCALE EXPANSION AND HOMOGENIZATION OF THE GRADIENT OF THE GREEN’S MATRIX 127 Substep 1.3.
In this substep, we study the term pertaining to the charges in the identity (3.8). We provethe expansion(3.21) 12 β ∆ H δ, ⋅ k + ∑ q ∈Q ∇ ∗ q ⋅ a q ∇ q H δ, ⋅ k = a β ∆ G δ, ⋅ k + ∑ i,j β ∇ i G δ,jk ∆ χ m,ij + ∑ i,j ∑ q ∈Q ∇ i G δ,jk ∇ ∗ q ⋅ a q ∇ q χ m,ij + R Q . where R Q is an error term which satisfies the H − ( B R + δ , µ β ) -norm estimate(3.22) ∥ R Q ∥ H − ( B R + δ µ β ) ≤ CR d − + γ α . We first compute the gradient and the Laplacian of the two-scale expansion H δ, ⋅ k using the notation of (1.6)of Chapter 2. We obtain the formulas(3.23) ∇H δ, ⋅ k = ∇ G δ, ⋅ k + ∑ i,j [∇∇ i G δ,jk ⊗ χ m,ij + ∇ i G δ,jk ∇ χ m,ij ] , and(3.24) ∆ H δ, ⋅ k = ∆ G δ, ⋅ k + ∑ i,j ∇ ⋅ (∇∇ i G δ,jk ⊗ χ m,ij ) + (∇∇ i G δ,jk ) ⋅ (∇ χ m,ij ) + (∇ i G δ,jk ) ∆ χ m,ij . We first treat the term ∆ H δ, ⋅ k and use the two following ingredients:(i) First we introduce the notation R Q, ∶= ∑ i,j ∇ ⋅ (∇∇ i G δ,jk ⊗ χ m,ij ) . By using the regularity esti-mate (3.4) on the homogenized Green’s matrix and the quantitative sublinearity of the correctorstated in Proposition 4.3 of Chapter 6, we prove that this term is an error term and estimate its H − ( B R + δ , µ β ) -norm according to the following computation ∥ R Q, ∥ H − ( B R + δ ,µ β ) ≤ C XXXXXXXXXXX∑ i,j ∇∇ i G δ,jk ⊗ χ m,ij XXXXXXXXXXX L ( B R + δ ,µ β ) ≤ ∑ i,j C ∥∇∇ i G δ,jk ∥ L ∞ ( B R + δ ,µ β ) ∥ χ m,ij ∥ L ( B R + δ ,µ β ) ≤ CR ( + δ )( − α ) R ( − δ )( d ) ≤ CR d − + γ α , where we have set γ α ∶= α ( + δ ) − δ ( d + ) > G δ, ⋅ k = ∇ ⋅ ∇ G δ, ⋅ k = ∑ i,j ∇ ⋅ (∇ i G δ,jk e ij ) .We obtain(3.25) ∆ H δ, ⋅ k = ∇ ⋅ ⎛⎝∑ i,j ∇ i G δ,jk ( e ij + ∇ χ m,ij )⎞⎠ + ∑ i,j (∇ i G δ,jk ) ∆ χ m,ij + R Q, . We then treat the term pertaining to the charges; the objective is to prove the identity(3.26) ∑ q ∈Q ∇ ∗ q ⋅ a q ∇ q H δ, ⋅ k = ∑ i,j ∇∇ i G δ,jk ∑ q ∈Q a q ∇ q ( l e ij + χ m,ij ) L t , d ∗ ( n q ) + ∑ q ∈Q ∇ i G δ,jk ∇ ∗ q ⋅ a q ∇ q ( l e ij + χ m,ij ) + R Q, , where R Q, is an error term which satisfies the estimate(3.27) ∥ R Q, ∥ H − ( B R + δ ,µ β ) ≤ CR d − − γ α . To prove this result, we select a test function h ∶ Z d → R ( d ) which belongs to the space H ( B R + δ , µ β ) andsatisfies the estimate ∥ h ∥ H ( B R + δ ,µ β ) ≤
1. For each charge q ∈ Q , we select a point x q which belongs to thesupport of the charge q arbitrarily. We then write(3.28) ∑ q ∈Q a q ∇ q H δ, ⋅ k ∇ q h = ∑ q ∈Q a q ( n q , d ∗ H δ, ⋅ k ) ( n q , d ∗ h ) .
28 7. QUANTITATIVE HOMOGENIZATION OF THE GREEN’S MATRIX
We use the exact formula for H δ and apply the codifferential. We obtaind ∗ H δ, ⋅ k = L t , d ∗ (∇H δ, ⋅ k ) = L , d ∗ ⎛⎝∇ G δ, ⋅ k + ∑ i,j [∇∇ i G δ,jk ⊗ χ m,ij + ∇ i G δ,jk ∇ χ m,ij ]⎞⎠ (3.29) = d ∗ G δ, ⋅ k + ∑ i,j ∇ i G δ,jk d ∗ χ m,ij + ∑ i,j L , d ∗ (∇∇ i G δ,jk ⊗ χ m,ij ) . We record the following formulad ∗ G δ, ⋅ k = L , d ∗ (∇ G δ, ⋅ k ) = L , d ∗ ⎛⎝∑ i,j ∇ i G δ,jk e ij ⎞⎠ = L , d ∗ ⎛⎝∑ i,j ∇ i G δ,jk ∇ l e ij ⎞⎠ (3.30) = ∑ i,j ∇ i G δ,jk L , d ∗ (∇ l e ij )= ∑ i,j ∇ i G δ,jk d ∗ l e ij . Putting the identities (3.29) and (3.30) back into (3.28), we obtain(3.31) ∑ q ∈Q a q ∇ q H δ, ⋅ k ∇ q h = ∑ i,j ∑ q ∈Q a q ( n q , ∇ i G δ,jk ( d ∗ l e ij + d ∗ χ m,ij )) ( n q , d ∗ h )·„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„¶ (3.31) −( i ) + ∑ q ∈Q a q ( n q , L , d ∗ (∇∇ i G δ,jk ⊗ χ m,ij )) ( n q , d ∗ h )·„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„¶ (3.31) −( ii ) . The second term (3.31)-(ii) is an error term which is small can be estimated thanks to the regularityestimate (3.4) and Young’s inequality. We obtain
RRRRRRRRRRRRR⟨ ∑ q ∈Q a q ( n q , L , d ∗ (∇∇ i G δ,jk ⊗ χ m,ij )) ( n q , d ∗ h )⟩ µ β RRRRRRRRRRRRR≤ ∑ q ∈Q e − c √ β ∥ q ∥ ∥ n q ∥ ∥∇ G δ,jk ∥ L ∞ ( Z d ,µ β ) ∥ χ m,ij ∥ L ( supp n q ,µ β ) ∥∇ h ∥ L ( supp n q ,µ β ) ≤ CR ( − δ )( d + ) ∑ q ∈Q e − c √ β ∥ q ∥ ∥ n q ∥ ∥ χ m,ij ∥ L ( supp n q ,µ β ) ∥∇ h ∥ L ( supp n q ,µ β ) ≤ CR ( − δ )( d + ) ∑ q ∈Q e − c √ β ∥ q ∥ ∥ n q ∥ ( R − + α ∥ χ m,ij ∥ L ( supp n q ,µ β ) + R − α ∥∇ h ∥ L ( supp n q ,µ β ) ) . We then use the inequality, for each point x ∈ Z d ,(3.32) ∑ q ∈Q e − c √ β ∥ q ∥ ∥ n q ∥ { x ∈ supp n q } ≤ C. We deduce that
RRRRRRRRRRRRR⟨ ∑ q ∈Q a q ( n q , L , d ∗ (∇∇ i G δ,jk ⊗ χ m,ij )) ( n q , d ∗ h )⟩ µ β RRRRRRRRRRRRR≤ CR − + α R ( − δ )( d + ) ∥ χ m,ij ∥ L ( B R + δ,µβ ) + CR − α R ( − δ )( d + ) ∥∇ h ∥ L ( B R + δ,µβ ) . We then use Proposition 4.3 of Chapter 6 and the assumption ∥∇ h ∥ L ( B R + δ,µβ ) ≤
1. We obtain(3.33)
RRRRRRRRRRRRR R ( + δ ) d ⟨ ∑ q ∈Q a q ( n q , L , d ∗ (∇∇ i G δ,jk ⊗ χ m,ij )) ( n q , d ∗ h )⟩ µ β RRRRRRRRRRRRR ≤ CR − α R ( − δ )( d + ) ≤ CR d − + γ α , where we have set γ α = α − δ ( d + ) . . TWO-SCALE EXPANSION AND HOMOGENIZATION OF THE GRADIENT OF THE GREEN’S MATRIX 129 To treat the term (3.31)-(i), we make use of the point x q and write ∑ q ∈Q a q ( n q , ∇ i G δ,jk ( d ∗ l e ij + d ∗ χ m,ij )) ( n q , d ∗ h )= ∑ q ∈Q a q ( n q , ( d ∗ l e ij + d ∗ χ m,ij )) ( n q , ∇ i G δ,jk d ∗ h )+ ∑ q ∈Q a q ( n q , (∇ i G δ,jk − ∇ i G δ,jk ( x q )) ( d ∗ l e ij + d ∗ χ m,ij )) ( n q , d ∗ h )+ ∑ q ∈Q a q ( n q ( d ∗ l e ij + d ∗ χ m,ij )) ( n q , (∇ i G δ,jk − ∇ i G δ,jk ( x q )) d ∗ h ) . The terms on the second and third lines are error terms which are small, they can be estimated by the regularityestimate (3.4) on the gradient of the homogenized Green’s matrix and Young’s inequality. We obtain
RRRRRRRRRRRRR⟨ ∑ q ∈Q a q ( n q , (∇ i G δ,jk − ∇ i G δ,jk ( x q )) ( d ∗ l e ij + d ∗ χ m,ij )) ( n q , d ∗ h )⟩ µ β RRRRRRRRRRRRR≤ ∑ q ∈Q e − c √ β ∥ q ∥ ∥ n q ∥ ∥∇ G δ,jk − ∇ G δ,jk ( x q )∥ L ∞ ( supp n q ,µ β ) ∥∇ χ m,ij ∥ L ( supp n q ,µ β ) ∥∇ h ∥ L ( supp n q ,µ β ) ≤ CR ( − δ ) d ∑ q ∈Q e − c √ β ∥ q ∥ ∥ n q ∥ (∥∇ χ m,ij ∥ L ( supp n q ,µ β ) + ∥∇ h ∥ L ( supp n q ,µ β ) ) . We then apply the estimate (3.32), the bound ∥∇ χ m,ij ∥ L ( B R + δ ,µ β ) ≤ C on the gradient of the corrector andthe assumption ∥∇ h ∥ L ( B R + δ ,µ β ) ≤ RRRRRRRRRRRRR R ( + δ ) d ⟨ ∑ q ∈Q a q ( n q , (∇ i G δ,jk − ∇ i G δ,jk ( x q )) ( d ∗ l e ij + d ∗ χ m,ij )) ( n q , d ∗ h )⟩ µ β RRRRRRRRRRRRR ≤ CR ( − δ ) d ≤ CR ( d − )+ γ , where we have set γ = − δ >
0. The same argument proves the inequality(3.35)
RRRRRRRRRRRRR R ( + δ ) d ⟨ ∑ q ∈Q a q ( n q ( d ∗ l e ij + d ∗ χ m,ij )) ( n q , (∇ i G δ,jk − ∇ i G δ,jk ( x q )) d ∗ h )⟩ µ β RRRRRRRRRRRRR ≤ CR ( d − )+ γ , with the same exponent γ >
0. Combining the identity (3.31) with the estimates (3.33), (3.34), (3.35), wehave obtained the following result: for each function h ∈ H ( B R + δ , µ β ) such that ∥ h ∥ H ( B R + δ ,µ β ) ≤
1, one hasthe estimate1 R ( + δ ) d ∑ q ∈Q a q ∇ q H δ, ⋅ k ∇ q h = R ( + δ ) d ∑ q ∈Q a q ( n q , ( d ∗ l e ij + d ∗ χ m,ij )) ( n q , ∇ i G δ,jk d ∗ h ) + O ( CR d − + γ α ) . We then use the identity ∇ i G δ,jk d ∗ h = d ∗ (∇ i G δ,jk h ) − L , d ∗ (∇∇ i G δ,jk ⊗ h ) which is established in (3.30). Wededuce that(3.36) ∑ q ∈Q a q ∇ q H δ, ⋅ k ∇ q h = ∑ q ∈Q a q ( n q , ( d ∗ l e ij + d ∗ χ m,ij )) ( n q , d ∗ (∇ i G δ,jk h ))+ ∑ q ∈Q a q ( n q , ( d ∗ l e ij + d ∗ χ m,ij )) ( n q , L , d ∗ (∇∇ i G δ,jk ⊗ h )) + O ( CR d − + γ α ) . This implies the identity (3.26) and the estimate (3.27).We now complete the proof of (3.21). To prove this identity, it is sufficient, in view of (3.25) and (3.26),to prove the estimate(3.37)12 β ∑ i,j (∇∇ i G δ,jk ) ⋅ ( e ij + ∇ χ m,ij ) + ∑ i,j (∇∇ i G δ,jk ) ∑ q ∈Q a q ∇ q ( l e ij + χ m,ij ) L t , d ∗ ( n q ) = ∇ ⋅ ( a β ∇ G δ, ⋅ k ) + R Q, , where the term R Q, satisfies the estimate(3.38) ∥ R Q, ∥ H − ( B R + δ ,µ β ) ≤ CR − d + γ α .
30 7. QUANTITATIVE HOMOGENIZATION OF THE GREEN’S MATRIX
The proof relies on the quantitative estimate for the H − ( B R + δ , µ β ) -norm of the flux corrector stated inProposition 4.3 of Chapter 6 and the regularity estimate (3.4) on the homogenized matrix G δ . We select afunction h ∶ Z d → R ( d ) which belongs to the space H ( B R + δ , µ β ) and such that ∥ h ∥ H ( B R + δ ,µ β ) ≤
1. We useit as a test function and write1 R ( + δ ) d RRRRRRRRRRRR⟨ ∑ x ∈ B R + δ ∑ i,j β (∇∇ i G δ,jk ( x, ⋅)) ⋅ ( e ij + ∇ χ m,ij ( x, ⋅)) h ( x, ⋅) (3.39) + ∑ i,j ∑ q ∈Q a q ∇ q ( l e ij + χ m,ij ) ( n q , L , d ∗ (∇∇ i G δ,jk ⊗ h )) − ∑ x ∈ B R + δ d ∑ i = ∇ ⋅ ( a β ∇ G δ, ⋅ k ( x )) h ( x, ⋅)⟩ µ β RRRRRRRRRRRRRR≤ C ∑ i,j XXXXXXXXXXX β ( e ij + ∇ χ m,ij ) + ∑ q ∈Q a q ∇ q ( l e ij + χ m,ij ) L t , d ∗ ( n q ) − a β e ij XXXXXXXXXXX H − ( B R + δ ,µ β ) ∥∇∇ i G δ,jk ⊗ h ∥ H ( B R + δ ,µ β ) . We then use Proposition 4.3 of Chapter 6 to write(3.40) ∑ i,j XXXXXXXXXXX β ( e ij + ∇ χ m,ij ) + ∑ q ∈Q a q ∇ q ( l e ij + χ n,i ) L t , d ∗ ( n q ) − a β e ij XXXXXXXXXXX H − ( B R + δ ,µ β ) ≤ CR ( + δ )( − α ) , and the regularity estimate (3.4) to write ∥∇∇ i G δ,jk h ∥ H ( B R + δ ,µ β ) ≤ R + δ ∥∇ G δ, ⋅ k ∥ L ∞ ( B R + δ ,µ β ) ∥ h ∥ L ( B R + δ ,µ β ) + ∥∇ G δ, ⋅ k ∥ L ∞ ( B R + δ ,µ β ) ∥ h ∥ L ( B R + δ ,µ β ) (3.41) + ∥∇ G δ, ⋅ k ∥ L ∞ ( B R + δ ,µ β ) ∥∇ h ∥ L ( B R + δ ,µ β ) ≤ C ( R d ( − δ ) + R ( d − )( − δ ) + R + δ R ( d + )( − δ ) )≤ R d − δ ( d + ) . Combining the estimates (3.39), (3.40) and (3.41), we have obtained that, for each function h ∈ H ( B R + δ , µ β ) such that ∥ h ∥ H ( B R + δ ,µ β ) ≤ R ( + δ ) d RRRRRRRRRRRR⟨ ∑ x ∈ B R + δ ∑ i,j (∇∇ i G δ,jk ( x )) ⋅ ( e ij + ∇ χ m,ij ( x, ⋅)) (3.42) + ∑ i,j ∑ q ∈Q a q ∇ q ( l e ij + χ m,ij ) ( n q , L , d ∗ (∇∇ i G δ,jk ⊗ h )) − ∑ x ∈ B R + δ a ∆ G δ, ⋅ k ( x ) ⋅ h ( x, ⋅)⟩ µ β RRRRRRRRRRRRRR≤ CR ( + δ )( − α ) R d − δ ( d + ) ≤ CR d − + γ α , where we have set γ ∶= α ( + δ ) − δ ( d + ) . Since the inequality (3.42) is valid for any function h ∈ H ( B R + δ , µ β ) satisfying ∥ h ∥ H ( B R + δ ,µ β ) ≤
1, the estimate (3.42) is equivalent to the identity (3.37) and the H − ( B R + δ , µ β ) -norm estimate (3.38). The proof of (3.37), and thus of (3.21), is complete. Substep 1.4
In this substep, we conclude Step 1 and prove the estimate (3.7). We use the identity (3.8)and the identities (3.9) proved in Substep 1, (3.10) proved in Substep 2 and (3.21) proved in Substep 3. We . TWO-SCALE EXPANSION AND HOMOGENIZATION OF THE GRADIENT OF THE GREEN’S MATRIX 131 obtain LH δ, ⋅ k = ∇ ⋅ a β ∇ G δ, ⋅ k (3.43) + ∑ i,j ∇ i G δ,jk ⎛⎝ ∆ φ χ m,ij + β ∆ χ m,ij + ∑ q ∈Q ∇ ∗ q ⋅ a q ∇ q ( l e ij + χ m,ij ) + β ∑ n ≥ β n (− ∆ ) n + χ m,ij ⎞⎠+ R Q + R ∆ n . We then treat the three lines of the previous display separately. For the first line, we use the identity(3.44) − ∇ ⋅ a β ∇ G δ, ⋅ k = ρ δ, ⋅ k in Z d . For the second line, we use that, by the definition of the finite-volume corrector given in Definition 4.1 ofChapter 6, this map is a solution of the Helffer-Sj¨ostrand equation
L ( l e ij + χ m,ij ) = B R + δ × Ω.We obtain ∑ i,j ∇ i G δ,jk ⎛⎝ ∆ φ χ m,ij + β ∆ χ m,ij + ∑ q ∈Q ∇ ∗ q ⋅ a q ∇ q ( l e ij + χ m,ij ) + β ∑ n ≥ β n (− ∆ ) n + χ m,ij ⎞⎠ = ∑ i,j ∇ i G δ,jk L ( l e ij + χ m,ij ) (3.45) = . For the third line, we use the estimates (3.11) and (3.22) on the error terms R Q and R ∆ n respectively. Weobtain(3.46) ∥ R Q + R ∆ n ∥ H − ( B R + δ ,µ β ) ≤ CR d − + γ α . A combination of the identities (3.43), (3.44), (3.45) and the estimate (3.46) proves the inequality ∥LH δ, ⋅ k − ρ δ, ⋅ k ∥ H − ( B R + δ ,µ β ) ≤ CR d − + γ α . The proof of the estimate (3.7) is complete. L -norm of the term ∇G δ − ∇H δ . The objective of this section is to prove thatthe gradient of the Green’s matrix ∇G δ and the gradient of the two-scale expansion ∇H δ are close in the L ( A R , µ β ) -norm. More specifically, we prove that there exists an exponent γ δ > ∥∇G δ − ∇H δ ∥ L ( A R ,µ β ) ≤ CR ( d − )+ γ δ . To prove this inequality, we work on the larger set B R + δ / and prove the estimate ∥∇G δ − ∇H δ ∥ L ( B R + δ / ,µ β ) ≤ CR ( + δ )( d − − ε / ) . (3.48)The inequality (3.47) implies (3.48); indeed by using that the annulus A R is included in the ball B R + δ we cancompute ∥∇G δ − ∇H δ ∥ L ( A R ,µ β ) ≤ ⎛⎝ ∣ B R + δ / ∣∣ A R ∣ ⎞⎠ ∥∇G δ − ∇H δ ∥ L ( B R + δ / ,µ β ) ≤ C ( R d ( + δ ) R d ) CR ( + δ )( d − − ε / ) ≤ CR d − + γ δ , where we have set γ δ ∶= δ ( d − − ε / ) . We now focus on the proof of the estimate (3.48). The strategy isto first fix an integer k ∈ { , . . . , ( d )} and use the identity LG δ, ⋅ k = ρ δ, ⋅ k to rewrite the estimate (3.7) in thefollowing form(3.49) ∥L (H δ, ⋅ k − G δ, ⋅ k )∥ H − ( B R + δ ,µ β ) ≤ CR d − + γ α . We then use the function G δ, ⋅ k −H δ, ⋅ k as a test function in the definition of the H − -norm in the inequality (3.49)to obtain the H -estimate stated in (3.48), as described in the outline of the proof at the beginning of this
32 7. QUANTITATIVE HOMOGENIZATION OF THE GREEN’S MATRIX chapter. The overall strategy is relatively straightforward; however, one has to deal with the following technicaldifficulty. By definition of the H − -norm, one needs to use a function in H ( B R + δ , µ β ) as a test function; inparticular the function must be equal to 0 outside the ball B R + δ . This condition is not verified by the function G δ, ⋅ k − H δ, ⋅ k which is thus not a suitable test function. To overcome this issue, we introduce a cutoff function η ∶ Z d → R supported in the ball B R + δ which satisfies the properties(3.50) 0 ≤ η ≤ B R + δ , η = B R + δ , and ∀ k ∈ N , ∣∇ k η ∣ ≤ CR ( + δ ) k , and use the function η (G δ, ⋅ k − H δ, ⋅ k ) as a test function. The main difficulty is thus to treat the cutoff function.Nevertheless, this difficulty is similar to the one treated in the proof of the Caccioppoli inequality stated inProposition 1.1 of Chapter 5. We will thus omit some of the details of the argument and refer the reader tothe proof of the Caccioppoli inequality for the missing elements of the proof.We use the function η (G δ, ⋅ k − H δ, ⋅ k ) as a test function in the inequality (3.49). We obtain1 R ( + δ ) d ∑ x ∈ B R + δ ⟨ η (G δ, ⋅ k − H δ, ⋅ k ) L (G δ, ⋅ k − H δ, ⋅ k )⟩ µ β ≤ ∥L (G δ, ⋅ k − H δ, ⋅ k )∥ H − ( B R + δ ,µ β ) ∥ η (G δ, ⋅ k − H δ, ⋅ k )∥ H ( B R + δ ,µ β ) (3.51) ≤ CR d − + γ α ∥ η (G δ, ⋅ k − H δ, ⋅ k )∥ H ( B R + δ ,µ β ) . We then treat the terms in the left and right sides of the inequality (3.51) separately. Regarding the left side,we prove the estimate(3.52) ∥ η (G δ, ⋅ k − H δ, ⋅ k )∥ H ( B R + δ ,µ β ) ≤ CR d − − δd . The proof relies on the properties of the cutoff function η stated in (3.50), the regularity estimate on theGreen’s matrix stated in Proposition 3.1 of Chapter 5, the L ∞ -bound on the homogenized Green’s matrix G δ hom stated in (3.4) and the bounds on the corrector and its gradient recalled below ∥ χ m,ij ∥ L ( B R + δ ,µ β ) ≤ CR ( + δ )( − α ) , ∥∇ χ m,ij ∥ L ( B R + δ ,µ β ) ≤ C and ∑ x ∈ Z d ∥ ∂ x χ m,ij ∥ L ( B R + δ ,µ β ) ≤ C. We first write(3.53) ∥ η (G δ, ⋅ k − H δ, ⋅ k )∥ H ( B R + δ ,µ β ) ≤ R + δ ∥ η (G δ, ⋅ k − H δ, ⋅ k )∥ L ( B R + δ ,µ β ) ·„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„¶ (3.53) −( i ) + ∥∇ η (G δ, ⋅ k − H δ, ⋅ k )∥ L ( B R + δ ,µ β ) ·„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„¶ (3.53) −( ii ) + ∥ η (∇G δ, ⋅ k − ∇H δ, ⋅ k )∥ L ( B R + δ ,µ β ) ·„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„¶ (3.53) −( iii ) + β ∑ x ∈ Z d ∥ η ( ∂ x G δ, ⋅ k − ∂ x H δ, ⋅ k )∥ L ( B R + δ ,µ β ) ·„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„¶ (3.53) −( iv ) , and treats the four terms in the right side separately. For the term (3.53)-(i), we use that the function η isnon-negative and smaller than 1 to write1 R + δ ∥ η (G δ, ⋅ k − H δ, ⋅ k )∥ L ( B R + δ ,µ β ) ≤ R + δ (∥G δ, ⋅ k ∥ L ( B R + δ ,µ β ) + ∥H δ, ⋅ k ∥ L ( B R + δ ,µ β ) ) . We then estimate the L -norm of the Green’s matrix G δ thanks to the estimate ∥G δ, ⋅ k ∥ L ( B R + δ ,µ β ) ≤ ∥G δ, ⋅ k ∥ L ∞ ( Z d ,µ β ) ≤ CR ( − δ )( d − ) . The L -norm of the two-scale expansion H can be estimated according to the following computation ∥H δ, ⋅ k ∥ L ( B R + δ ,µ β ) ≤ ∥ G δ, ⋅ k ∥ L ( B R + δ ,µ β ) + ∑ i,j ∥∇ i G δ,jk ∥ L ∞ ( B R + δ ,µ β ) ∥ χ m,ij ∥ L ( B R + δ ,µ β ) ≤ CR ( − δ )( d − ) + CR ( + δ )( − α ) R ( − δ )( d − ) ≤ CR ( − δ )( d − ) , . TWO-SCALE EXPANSION AND HOMOGENIZATION OF THE GRADIENT OF THE GREEN’S MATRIX 133 where we have used the inequality α ≫ δ in the third inequality. A combination of the three previous displaysshows the estimate(3.54) 1 R + δ ∥ η (G δ, ⋅ k − H δ, ⋅ k )∥ L ( B R + δ ,µ β ) ≤ CR + δ × R ( − δ )( d − ) ≤ CR d − − δ ( d − ) . The proof of the term (3.53)-(ii) is identical, we use the estimate ∣∇ η ∣ ≤ CR + δ and apply the estimate obtainedfor the term (3.53)-(ii). We obtain(3.55) ∥∇ η (G δ, ⋅ k − H δ, ⋅ k )∥ L ( B R + δ ,µ β ) ≤ CR d − − δ ( d − ) . For the term (3.53)-(iii), we first write ∥ η (∇G δ, ⋅ k − ∇H δ, ⋅ k )∥ L ( B R + δ ,µ β ) ≤ ∥∇G δ, ⋅ k ∥ L ∞ ( Z d ,µ β ) + ∥∇H δ, ⋅ k ∥ L ( B R + δ ,µ β ) . The L ∞ -norm of the Green’s matrix ∇G δ, ⋅ k is estimated by Proposition 3.1. We have ∥∇G δ, ⋅ k ∥ L ∞ ( Z d ,µ β ) ≤ CR ( − δ )( d − − ε ) . For the L -norm of the two-scale expansion H , we use the formula (3.23) and write ∥∇H δ, ⋅ k ∥ L ( B R + δ ,µ β ) ≤ ∥∇ G δ, ⋅ k ∥ L ∞ ( Z d ) + ∑ i,j ∥∇∇ i G δ,jk ∥ L ∞ ( Z d ) ∥ χ m,ij ∥ L ( B R + δ ,µ β ) + ∑ i,j ∥∇ i G δ,jk ∥ L ∞ ( Z d ) ∥∇ χ m,ij ∥ L ( B R + δ ,µ β ) ≤ CR ( − δ )( d − − ε ) + CR ( + δ )( − α ) R ( − δ )( d − ε ) + CR ( − δ )( d − − ε ) ≤ CR d − − ε − δ ( d − − ε ) . A combination of the three previous displays together with the inequality δ ≫ ε yields the estimate(3.56) ∥ η (∇G δ, ⋅ k − ∇H δ, ⋅ k )∥ L ( B R + δ ,µ β ) ≤ CR d − − δd . There remains to estimate the term (3.53)-(iv). We first write(3.57) β ∑ x ∈ Z d ∥ η ( ∂ x G δ, ⋅ k − ∂ x H δ, ⋅ k )∥ L ( B R + δ ,µ β ) ≤ β ∑ x ∈ Z d ∥ η∂ x G δ, ⋅ k ∥ L ( B R + δ ,µ β ) ·„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„¶ (3.57) −( i ) + β ∑ x ∈ Z d ∥ η∂ x H δ ∥ L ( B R + δ ,µ β ) ·„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„¶ (3.57) −( ii ) and estimate the two terms in the right side separately. For the term (3.57)-(i), we use that the map G δ solvesthe equation LG δ, ⋅ k = ρ δ and use the map η G δ as a test function. We obtain β ∑ x ∈ Z d ∥ η∂ x G δ, ⋅ k ∥ L ( B R + δ ,µ β ) = − ∑ x ∈ Z d ⟨∇G δ, ⋅ k ( x, ⋅) ⋅ ∇ ( η G δ, ⋅ k ) ( x, ⋅)⟩ µ β − β ∑ q ∈Q ⟨∇ q G δ, ⋅ k ⋅ a q ∇ q ( η G δ, ⋅ k )⟩ µ β − ∑ n ≥ ∑ x ∈ Z d β n ⟨∇ n + G δ, ⋅ k ( x, ⋅) ⋅ ∇ n + ( η G δ, ⋅ k ) ( x, ⋅)⟩ µ β + β ∑ x ∈ Z d ρ δ, ⋅ k ( x ) η ( x ) ⋅ ⟨G δ, ⋅ k ( x, ⋅)⟩ µ β . We then estimate the four terms in the right sides using the pointwise estimates on the function G δ andits gradient stated in Proposition 3.1, the properties on the functions ρ δ and η stated in (2.4) and (3.50)respectively. We omit the technical details and obtain the estimate(3.58) ∑ x ∈ Z d ∥ η∂ x G δ, ⋅ k ∥ L ( B R + δ ,µ β ) ≤ CR ( − δ )( d − − ε ) . The term (3.57)-(ii) involving the two-scale expansion is the easiest one to estimate; using the explicitformula for the map H δ, ⋅ k and the fact that the function G δ does not depend on the field φ , we have theidentity ∂ x H δ, ⋅ k ∶= ∑ i,j ∇ i G δ,jk ∂ x χ m,ij .
34 7. QUANTITATIVE HOMOGENIZATION OF THE GREEN’S MATRIX
We deduce that ∑ x ∈ Z d ∥ η∂ x H δ, ⋅ k ∥ L ( B R + δ ,µ β ) ≤ ∑ x ∈ Z d ∥ ∂ x H δ, ⋅ k ∥ L ( B R + δ ,µ β ) (3.59) ≤ C ∑ i,j ∥∇ G δ, ⋅ k ∥ L ∞( Z d ) ∑ x ∈ Z d ∥ ∂ x χ m,ij ∥ L ( B R + δ ,µ β ) ≤ CR ( − δ )( d − − ε ) . Combining the inequalities (3.57), (3.58) and (3.59) yields(3.60) ∑ x ∈ Z d ∥ η ( ∂ x G δ, ⋅ k − ∂ x H δ, ⋅ k )∥ L ( B R + δ ,µ β ) ≤ CR d − − ε − δ ( d − − ε ) ≤ CR d − − δd . The inequality (3.52) is then obtained by combining the estimates (3.54), (3.55), (3.56) and (3.60). Wethen put the inequality back into the inequality (3.51) and deduce that(3.61) 1 R ( + δ ) d ∑ x ∈ B R + δ ⟨ η (G δ, ⋅ k − H δ, ⋅ k ) L (G δ, ⋅ k − H δ, ⋅ k )⟩ µ β ≤ CR d − + γ α × R d − − δd ≤ CR d − + γ α , where we have used in the second inequality that the exponent γ α is of order α ; it is thus much larger thanthe value δd and we may write γ α − δd = γ α following the conventional notation described at the beginning ofSection 3.In the rest of this step, we treat the left side of (3.61) and prove the inequality(3.62) ∥∇G δ, ⋅ k − ∇H δ, ⋅ k ∥ L ( B R + δ ,µ β ) ≤ R ( + δ ) d ∑ x ∈ B R + δ ⟨ η (G δ, ⋅ k − H δ, ⋅ k ) L (G δ, ⋅ k − H δ, ⋅ k )⟩ µ β + CR ( + δ )( d − − ε ) . First, by definition of the Helffer-Sj¨ostrand operator L , we have the identity ∑ x ∈ Z d ⟨ η (G δ, ⋅ k − H δ, ⋅ k ) L (G δ, ⋅ k − H δ, ⋅ k )⟩ µ β (3.63) = ∑ x,y ∈ Z d η ( x ) ⟨( ∂ y G δ, ⋅ k ( x, ⋅) − ∂ y H δ, ⋅ k ( x, ⋅)) ⟩ µ β (3.64) + β ∑ x ∈ Z d ⟨(∇G δ, ⋅ k − ∇H δ, ⋅ k ) ( x, ⋅) ⋅ ∇ ( η (G δ, ⋅ k − H δ, ⋅ k )) ( x, ⋅)⟩ µ β + ∑ q ∈Q ⟨∇ q (G δ, ⋅ k − H δ, ⋅ k ) ⋅ a q ∇ q ( η (G δ, ⋅ k − H δ, ⋅ k ))⟩ µ β + β ∑ n ≥ ∑ x ∈ Z d β n ⟨∇ n + (G δ, ⋅ k − H δ, ⋅ k ) ( x, ⋅) ⋅ ∇ n + ( η (G δ, ⋅ k − H δ, ⋅ k )) ( x, ⋅)⟩ µ β . We then estimate the four terms on the right side separately. For the first one, we use that it is non-negative ∑ x,y ∈ Z d η ( x ) ⟨( ∂ y G δ, ⋅ k ( x, ⋅) − ∂ y H δ, ⋅ k ( x, ⋅)) ⟩ µ β ≥ . For the second one, we expand the gradient of the product η (G δ, ⋅ k − H) and write ∑ x ∈ Z d ⟨(∇G δ, ⋅ k − ∇H δ, ⋅ k ) ( x, ⋅) ⋅ ∇ ( η (G δ, ⋅ k − H δ, ⋅ k )) ( x, ⋅)⟩ µ β (3.65) = ∑ x ∈ Z d η ( x ) ⟨(∇G δ, ⋅ k − ∇H δ ) ( x, ⋅) ⋅ ∇ (G δ, ⋅ k − H δ, ⋅ k ) ( x, ⋅)⟩ µ β + ∑ x ∈ Z d ⟨(∇G δ, ⋅ k − ∇H δ, ⋅ k ) ( x, ⋅) ⋅ ∇ η ( x ) (G δ, ⋅ k − H δ, ⋅ k ) ( x, ⋅)⟩ µ β . we divide the identity (3.65) by the volume factor R ( + δ ) d and use the properties of the function η statedin (3.50). In particular, we use that the gradient of η is supported in the annulus A R + δ ∶= B R + δ ∖ B R + δ andobtain(3.66) 1 R ( + δ ) d ∑ x ∈ Z d ⟨(∇G δ, ⋅ k − ∇H δ, ⋅ k ) ( x, ⋅) ⋅ ∇ ( η (G δ, ⋅ k − H δ, ⋅ k )) ( x, ⋅)⟩ µ β ≥ c ∥ η (∇G δ, ⋅ k − ∇H δ, ⋅ k )∥ L ( B R + δ ,µ β ) − CR + δ ∥∇G δ, ⋅ k − ∇H δ, ⋅ k ∥ L ( A R + δ ,µ β ) ∥G δ, ⋅ k − H δ, ⋅ k ∥ L ( A R + δ ,µ β ) . . TWO-SCALE EXPANSION AND HOMOGENIZATION OF THE GRADIENT OF THE GREEN’S MATRIX 135 By a computation similar to the one performed for the term (3.53)-(iii), but using the estimates (3.1) for theGreen’s matrices in the distant annulus A R + δ , instead of the L ∞ -estimates (3.2) and (3.4). We obtain(3.67) ∥∇G δ, ⋅ k − ∇H δ, ⋅ k ∥ L ( A R + δ ,µ β ) ≤ CR ( + δ )( d − − ε ) and ∥G δ, ⋅ k − H δ, ⋅ k ∥ L ( A R + δ ,µ β ) ≤ CR ( + δ )( d − ) . A combination of the inequalities (3.66) and (3.67) proves the estimate(3.68) 1 R ( + δ ) d ∑ x ∈ Z d ⟨(∇G δ, ⋅ k − ∇H δ, ⋅ k ) ( x, ⋅) ⋅ ∇ ( η (G δ − H δ, ⋅ k )) ( x, ⋅)⟩ µ β + CR ( + δ )( d − − ε ) ≥ c ∥ η (∇G δ, ⋅ k − ∇H δ, ⋅ k )∥ L ( B R + δ ,µ β ) . The other terms in the right side of the identity (3.63) involving the sum over the iteration of the Laplacianand over the charges q ∈ Q are treated similarly. Instead of expanding the gradient as it was done in (3.65), weuse the commutation estimates (see (1.8) and (1.21) in the proof of Proposition 1.1 in Chapter 5), for eachinteger n ∈ N , and each pair ( x, φ ) ∈ Z d × Ω,(3.69) ∣∇ n ( η (G δ, ⋅ k − H δ, ⋅ k )) ( x, φ ) − η ( x )∇ n (G δ, ⋅ k − H δ, ⋅ k ) ( x, φ )∣ ≤ C n R + δ ∑ z ∈ B ( x,n ) ∣G δ, ⋅ k ( z, φ ) − H δ, ⋅ k ( z, φ )∣ . and for each charge q ∈ Q , each point x in the support of the charge q and each field φ ∈ Ω, ∣∇ q ( η (G δ, ⋅ k − H δ, ⋅ k )) − η ( x )∇ q (G δ, ⋅ k − H δ, ⋅ k )∣ ≤ CR + δ ( diam q ) ∥ q ∥ ∑ z ∈ supp q ∣G δ, ⋅ k − H∣ . The details of the argument are similar to the one presented in the proof of the Caccioppoli inequality(Proposition 1.1 of Chapter 5) and are omitted. The results obtained are stated below(3.70)1 R ( + δ ) d ∑ q ∈Q ⟨∇ q (G δ, ⋅ k − H δ, ⋅ k ) ⋅ a q ∇ q ( η (G δ, ⋅ k − H δ, ⋅ k ))⟩ µ β + CR ( + δ )( d − − ε ) ≥ − Ce − c √ β ∥ η (∇G δ, ⋅ k − ∇H δ, ⋅ k )∥ L ( Z d ,µ β ) and 1 R ( + δ ) d ∑ n ≥ ∑ x ∈ Z d β n ⟨∇ n + (G δ, ⋅ k − H δ, ⋅ k ) ( x, ⋅) ⋅ ∇ n + ( η (G δ, ⋅ k − H δ, ⋅ k )) ( x, ⋅)⟩ µ β + CR ( + δ )( d − − ε ) (3.71) ≥ ∑ n ≥ ∑ x ∈ Z d β n ⟨ η ( x ) ∣∇ n + (G δ, ⋅ k − H δ, ⋅ k ) ( x, ⋅)∣ ⟩ µ β ≥ . We then combine the identity (3.63) with the estimates (3.68), (3.70) and (3.71) and assume that the inversetemperature β is large enough. We obtain1 R ( + δ ) d ∑ x ∈ Z d ⟨ η (G δ, ⋅ k − H) L (G δ, ⋅ k − H δ, ⋅ k )⟩ µ β + CR ( + δ )( d − − ε ) ≥ c ∥ η (∇G δ, ⋅ k − ∇H δ, ⋅ k )∥ L ( Z d ,µ β ) ≥ c ∥∇G δ, ⋅ k − ∇H δ, ⋅ k ∥ L ( B R + δ / ,µ β ) . The proof of the inequality (3.62) is then complete. To complete the proof of Step 2, we combine theestimates (3.61) and (3.62). We obtain(3.72) ∥∇G δ, ⋅ k − ∇H δ, ⋅ k ∥ L ( B R + δ ,µ β ) ≤ CR d − + γ α + CR ( + δ )( d − − ε ) ≤ CR ( + δ )( d − − ε ) , where the last inequality is a consequence of the fact that γ α is of order α and of the ordering α ≫ δ ≫ ε .Since the inequality (3.72) is valid for any integer k ∈ { , . . . , ( d )} , the proof of the estimate (3.48) is complete.
36 7. QUANTITATIVE HOMOGENIZATION OF THE GREEN’S MATRIX
In this section, we post-process theconclusion (3.48) of Section 3.3 and prove that the gradient of the Green’s matrix ∇G ⋅ k is close to the map ∑ i,j ( e ij + ∇ χ ij ) ∇ i G jk . The objective is to prove that there exists an exponent γ δ > XXXXXXXXXXX∇G ⋅ k − ∑ i,j ( e ij + ∇ χ ij ) ∇ i G jk XXXXXXXXXXX L ( A R ,µ β ) ≤ CR d − + γ δ . We first use the regularity estimates stated in Proposition 3.1 and the L -bound on the gradient of the infinite-volume corrector, for each x ∈ Z d , each pair of integers ( i, j ) ∈ { , . . . , d } × { , . . . , ( d )} , ∥∇ χ ij ( x, ⋅)∥ L ( µ β ) ≤ C .We write XXXXXXXXXXX∇ (G ⋅ k − G δ, ⋅ k ) − ∑ i,j ( e ij + ∇ χ ij ) ∇ i ( G δ,jk − G jk )XXXXXXXXXXX L ( A R ,µ β ) (3.74) ≤ ∥∇ (G ⋅ k − G δ, ⋅ k )∥ L ( A R ,µ β ) + ∑ i,j ∥( e ij + ∇ χ ij )∥ L ( A R ,µ β ) ∥∇ i ( G δ,jk − G jk )∥ L ∞ ( A R ,µ β ) ≤ CR d − + γ δ . Using the inequality (3.74), we see that to prove (3.73) it is sufficient to prove the estimate(3.75)
XXXXXXXXXXX∇G δ, ⋅ k − ∑ i,j ( e ij + ∇ χ ij ) ∇ i G δ,jk XXXXXXXXXXX L ( A R ,µ β ) ≤ CR d − + γ δ . We then use the main estimate (3.48) and deduce that, to prove the inequality (3.75), it is sufficient to prove(3.76)
XXXXXXXXXXX∇H δ, ⋅ k − ∑ i,j ( e ij + ∇ χ ij ) ∇ i G δ,jk XXXXXXXXXXX L ( A R ,µ β ) ≤ CR d − + γ δ . The rest of the argument of this step is devoted to the proof of (3.76). We first use the explicit formula for thegradient of the two-scale expansion ∇H δ, ⋅ k stated in (3.23) and write XXXXXXXXXXX∇H δ, ⋅ k − ∑ i,j ( e ij + ∇ χ m,ij ) ∇ i G δ,jk XXXXXXXXXXX L ( A R ,µ β ) ≤ ∑ i,j ∥∇∇ i G δ,jk χ m,ij ∥ L ( A R ,µ β ) + ∥(∇ i G δ,jk ) (∇ χ m,ij − ∇ χ ij )∥ L ( A R ,µ β ) . We then use the regularity estimate (3.4), the quantitative sublinearity of the corrector stated in Proposition 4.3and Proposition 4.4 of Chapter 6 to quantify the L -norm of the difference between the gradient of finite-volumecorrector and the gradient of the infinite-volume corrector. We obtain XXXXXXXXXXX∇H δ, ⋅ k − ∑ i,j ( e ij + ∇ χ m,ij ) ∇ i G δ,jk XXXXXXXXXXX L ( A R ,µ β ) ≤ ( ∣ A R ∣∣ B R + δ ∣ ) CR − α R d − ε + ( ∣ A R ∣∣ B R + δ ∣ ) CR − α R d − − ε (3.77) ≤ CR d − + γ α , where we have set γ α ∶= α − ε − dδ . Using that the exponent γ α is larger than the exponent γ δ completes theproof of the estimate (3.73).
4. Homogenization of the mixed derivative of the Green’s matrix
In this section, we use Proposition 2.1 to prove Theorem 4. We fix a charge q ∈ Q and recall the definitionsof the maps U q and G q given in the statement of Theorem 2.1. The proof is decomposed into four sectionsand follows the outline of the proof given in Section 2.2. . HOMOGENIZATION OF THE MIXED DERIVATIVE OF THE GREEN’S MATRIX 137 In this section, we record some properties pertaining to the functions U q and G q which are used in the argument. Proposition . There exists an inverse temperature β ∶= β ( d ) < and a constant C q which satisfiesthe estimate C q ≤ C ∥ q ∥ k , for some C ( d ) < ∞ and k ( d ) < ∞ , such that the following statement holds. Foreach pair of points y ∈ Z d and each integer k ∈ N , one has the estimates ∥∇U q ( y, ⋅)∥ L ∞ ( µ β ) ≤ C q ∣ y ∣ d − ε , ∥U q ( y, ⋅)∥ L ∞ ( µ β ) ≤ C q ∣ y ∣ d − − ε and ∣∇ k G q ( y )∣ ≤ C q ∣ y ∣ d − + k . Proof.
The proof is a consequence of the regularity estimates given in Proposition 4.7 of Chapter 3 andthe identity q = d n q . (cid:3) The objective of this section isto use Proposition 3.1 and the symmetry of the Helffer-Sj¨ostrand operator L to prove the following estimate ofthe expectations(4.1) ⎛⎝ R − d ∑ z ∈ A R ∣⟨U q ( z, ⋅)⟩ µ β − G q ( z )∣ ⎞⎠ ≤ CR d + γ δ . We start from the formula, for each integer k ∈ { , . . . , ( d )} (4.2) XXXXXXXXXXX d ∗ G ⋅ k − ∑ i,j ( d ∗ l e ij + d ∗ χ ij ) ∇ i G jk XXXXXXXXXXX L ( A R ,µ β ) ≤ CR d − + γ δ , which is a direct consequence of Proposition 2.1 since the codifferential is a linear functional of the gradient.Using the estimate (4.2), we deduce that R − d ∑ x ∈ A R RRRRRRRRRRR⟨ cos 2 π ( φ, q ( x + ⋅)) ( n q ( x + ⋅) , d ∗ G ⋅ k )⟩ µ β − ∑ i,j ⟨ cos 2 π ( φ, q ( x + ⋅)) ( n q ( x + ⋅) ( d ∗ l e ij + d ∗ χ ij ))⟩ µ β ∇ i G jk ( x )RRRRRRRRRRR≤ C q R d − + γ δ . By the translation invariance of the measure µ β and the stationarity of the gradient of the infinite-volumecorrector, we deduce that ∑ i,j ⟨ cos 2 π ( φ, q ( x + ⋅)) ( n q ( x + ⋅) ( d ∗ l e ij + d ∗ χ ij ))⟩ µ β ∇ i G jk ( x )= ∑ i,j ⟨ cos 2 π ( φ, q ) ( n q ( d ∗ l e ij + d ∗ χ ij ))⟩ µ β ∇ i G jk ( x )= G q ( x ) . We now claim that we have the identity, for each point x ∈ A R , ⟨ cos 2 π ( φ, q ( x + ⋅)) ( n q ( x + ⋅) , d ∗ G)⟩ µ β = ⟨U q ( x +⋅) ( , ⋅)⟩ µ β . The proof of this result is a consequence of the symmetry of the Helffer-Sj¨ostrand operator L . To argue this,we use the computation ⟨ cos 2 π ( φ, q ( x + ⋅)) ( n q ( x + ⋅) , d ∗ G)⟩ µ β = ⟨( cos 2 π ( φ, q ( x + ⋅)) q ( x + ⋅) , G)⟩ µ β = ⟨( cos 2 π ( φ, q ( x + ⋅)) q ( x + ⋅) , L − δ )⟩ µ β = ⟨(L − cos 2 π ( φ, q ( x + ⋅)) q ( x + ⋅) , δ )⟩ µ β = ⟨U q ( x +⋅) ( , ⋅)⟩ µ β . A combination of the four previous displays implies(4.3) R − d ∑ x ∈ A R ∣⟨U q ( x +⋅) ( , ⋅)⟩ µ β − G q ( x )∣ ≤ CR d + γ δ .
38 7. QUANTITATIVE HOMOGENIZATION OF THE GREEN’S MATRIX
We then use the translation invariance of the measure µ β and the definition of the map U q as the solution ofthe Helffer-Sj¨ostrand equation (1.2) to write(4.4) ⟨U q ( x +⋅) ( , ⋅)⟩ µ β = ⟨U q ( x, ⋅)⟩ µ β . Combining the inequality (4.3) with the identity (4.4), we obtain R − d ∑ x ∈ A R ∣⟨U q ( x, ⋅)⟩ µ β − G q ( x )∣ ≤ CR d + γ δ . We finally upgrade the L -inequality stated in (4.3) into an L -inequality: by using Proposition 4.1, we write R − d ∑ x ∈ A R ∣⟨U q ( x, ⋅)⟩ µ β − G q ( x )∣ ≤ ⎛⎝ R − d ∑ x ∈ A R ∣⟨U q ( x, ⋅)⟩ µ β − G q ( x )∣⎞⎠ (∥U q ( x, ⋅)∥ L ∞ ( A R ,µ β ) + ∥ G q ∥ L ∞ ( A R ) )≤ C q R d − + γ δ × R d − − ε ≤ C q R d − + γ δ , where we have used the convention notation described at the beginning of Section 3 to absorb the exponent ε into the exponent γ δ in the third inequality. U q . In this section, we prove that the random variable U q contracts around its expectation. To this end, we prove the variance estimate, for each point z ∈ Z d ,(4.5) var [U q ( z, ⋅)] ≤ C q ∣ z ∣ d − ε . Let us make a comment about the result: since the size of the random variable U q ( z, ⋅) is of order ∣ z ∣ − d (sinceit behaves like the gradient of a Green’s function), we would expect its variance to be of order ∣ z ∣ − d . Theinequality (4.5) asserts that it is in fact of order ∣ z ∣ d − ε which is smaller than the typical size of the randomvariable U q ( z, ⋅) by an algebraic factor.Once this estimate is established, we can combine it with the main result (4.1) of Section 4.2 to provethat the map U q is close to the (deterministic) Green’s function G q in the L ( A R , µ β ) -norm: we obtain theinequality(4.6) ∥U q − G q ∥ L ( A R ,µ β ) ≤ CR d − − γ δ . We now prove of the variance estimate (4.5). We first apply the Brascamp-Lieb inequality and write(4.7) var [U q ( z, ⋅)] ≤ C ∑ y,y ∈ Z d ∥ ∂ y U q ( z, ⋅)∥ L ( µ β ) C ∣ y − y ∣ d − ∥ ∂ y U q ( z, ⋅)∥ L ( µ β ) . A consequence of the inequality (4.7) is that to estimate the variance of the random variable U q ( z, ⋅) , itis sufficient to understand the behavior of the mapping y ↦ ∂ y U q ( z, ⋅) . To this end, we appeal to thedifferentiated Helffer-Sj¨ostrand equation: following the arguments developed in Section 4 of Chapter 5, themap u ∶ ( y, z, φ ) ↦ ∂ y U q ( z, φ ) is solution of the equation L der u ( x, y, φ ) = − ∑ q ∈Q πz ( β, q ) cos 2 π ( φ, q ) (U q , q ) q ( x )⊗ q ( y )+ π sin 2 π ( φ, q ) q ( x )⊗ q ( y ) in Z d × Z d × Ω . The function u can be expressed in terms of the Green’s function G der and we write, for each triplet ( x, y, φ ) ∈ Z d × Z d × Ω, u ( x, y, φ ) = ∑ q ∈Q πz ( β, q ) ∑ x ,y ∈ Z d d ∗ x d ∗ y G der , cos 2 π ( φ,q )(U q ,q ) ( x, y, φ ; x , y ) n q ( x ) ⊗ n q ( y )+ ∑ x ,y ∈ Z d π d ∗ x d ∗ y G der , sin 2 π ( φ,q ) ( x, y, φ ; x , y ) n q ( x ) ⊗ n q ( y ) . . HOMOGENIZATION OF THE MIXED DERIVATIVE OF THE GREEN’S MATRIX 139 We use the regularity estimates on the Green’s function stated in Proposition 4.2 of Chapter 5 to obtain, foreach pair of points ( x, y ) ∈ Z d × Z d , ∥ u ( x, y, ⋅)∥ L ∞ ( µ β ) ≤ C ∑ q ∈Q e − c √ β ∥ q ∥ ∑ x ,y ∈ Z d ∣ n q ( x )∣ ∣ n q ( y )∣ ∥( d ∗ U q , n q )∥ L ∞ ( µ β ) ∣ x − x ∣ d − ε + ∣ y − y ∣ d − ε ·„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„¶ (4.8) −( i ) (4.8) + ∑ x ,y ∈ Z d ∣ n q ( x )∣ ∣ n q ( y )∣∣ x − x ∣ d − ε + ∣ y − y ∣ d − ε ·„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„¶ (4.8) −( ii ) . We then estimate the two terms (4.8)-(i) and (4.8)-(ii) separately. We first focus on the term (4.8)-(i) andprove the inequality(4.9) (4.8) − ( i ) ≤ C q ∣ x − y ∣ d − ε max (∣ x ∣ , ∣ y ∣) d − To prove the estimate (4.9), we first decompose the set of charges Q according to the following procedure. Foreach z ∈ Z d , we denote by Q z the set of charges q ∈ Q such that the point z belongs to the support of n q , i.e., Q z ∶= { q ∈ Q ∶ z ∈ supp n q } . We note that we have the equality Q ∶= ⋃ z ∈ Z d Q z but the collection (Q z ) z ∈ Z d isnot a partition of Q . We first prove that, for each point z ∈ Z d ,(4.10) ∑ q ∈Q z e − c √ β ∥ q ∥ ∑ x ,y ∈ Z d ∣ n q ( x )∣ ∣ n q ( y )∣ ∥( d ∗ U q , n q )∥ L ∞ ( µ β ) ∣ x − x ∣ d − ε + ∣ y − y ∣ d − ε ≤ C q (∣ x − z ∣ d − ε + ∣ y − z ∣ d − ε ) × ∣ z ∣ d − ε To prove the estimate (4.10), we first use Proposition 4.1 to estimate the term ∥( d ∗ U q , n q )∥ L ∞ ( µ β ) . We write,for each charge q ∈ Q z , ∥( d ∗ U q , n q )∥ L ∞ ( µ β ) ≤ ∥∇U q ∥ L ∞ ( supp n q ,µ β ) ∥ n q ∥ L (4.11) ≤ C q ,q sup z ∈ supp n q ∣ z ∣ d − ε . Since we have assumed that the point z belongs to the support of the charge n q , we have the inequality(4.12) sup z ∈ supp n q ∣ z ∣ d − ε ≤ ∣ diam n q ∣ d − ε ∣ z ∣ d − ε ≤ C q ∣ z ∣ d − ε . Combining the estimates (4.11), (4.12), we deduce that(4.13) ∥( d ∗ U q , n q )∥ L ∞ ( µ β ) ≤ C q ,q ∣ z ∣ d − ε . Putting the inequality (4.13) back into the left side of the estimate (4.10), we obtain ∑ q ∈Q z e − c √ β ∥ q ∥ ∑ x ,y ∈ Z d ∣ n q ( x )∣ ∣ n q ( y )∣ ∥( d ∗ U q , n q )∥ L ∞ ( µ β ) ∣ x − x ∣ d − ε + ∣ y − y ∣ d − ε (4.14) ≤ ∑ q ∈Q z Ce − c √ β ∥ q ∥ C q,q ∣ z ∣ d − ε ∑ x ,y ∈ supp n q ∣ x − x ∣ d − ε + ∣ y − y ∣ d − ε . Using that, for a given charge q ∈ Q z , the point z belongs to the support of n q , we write(4.15) 1 ∣ x − x ∣ d − ε + ∣ y − y ∣ d − ε ≤ ∣ diam n q ∣ d ∣ x − z ∣ d − ε + ∣ y − z ∣ d − ε ≤ C q ∣ x − z ∣ d − ε + ∣ y − z ∣ d − ε . We then combine the estimates (4.14) and (4.15) and use the exponential decay of the coefficient e − c √ β ∥ q ∥ toabsorb the algebraic growth of the constant C q ,q in the parameter ∥ q ∥ . We obtain ∑ q ∈Q z e − c √ β ∥ q ∥ ∑ x ,y ∈ Z d ∣ n q ( x )∣ ∣ n q ( y )∣ ∥( d ∗ U q , n q )∥ L ∞ ( µ β ) ∣ x − x ∣ d − ε + ∣ y − y ∣ d − ε ≤ C q ∣ z ∣ d − ε × (∣ x − z ∣ d − ε + ∣ y − z ∣ d − ε ) .
40 7. QUANTITATIVE HOMOGENIZATION OF THE GREEN’S MATRIX
The proof of the estimate (4.10) is complete. We then use the identity
Q ∶= ⋃ z ∈ Z d Q z to write(4.8) − ( i ) = ∑ q ∈Q e − c √ β ∥ q ∥ ∑ x ,y ∈ Z d ∣ n q ( x )∣ ∣ n q ( y )∣ ∥( d ∗ U q , n q )∥ L ∞ ( µ β ) ∣ x − x ∣ d − ε + ∣ y − y ∣ d − ε (4.16) ≤ ∑ z ∈ Z d ∑ q ∈Q z e − c √ β ∥ q ∥ ∑ x ,y ∈ Z d ∣ n q ( x )∣ ∣ n q ( y )∣ ∥( d ∗ U q , n q )∥ L ∞ ( µ β ) ∣ x − x ∣ d − ε + ∣ y − y ∣ d − ε ≤ ∑ z ∈ Z d C q ∣ z ∣ d − ε × (∣ x − z ∣ d − ε + ∣ y − z ∣ d − ε ) . We estimate the sum in the right side of the estimate (4.16). To this end, we note that, for each triplet ( x, y, z ) ∈ Z d × Z d × Z d , (∣ x − z ∣ d − ε + ∣ y − z ∣ d − ε ) ≥ c dist Z d (( x, y ) , ( z, z )) d − ε ≥ c (∣ x − y ∣ + ∣ x + y − z ∣ ) d − ε (4.17) ≥ c (∣ x − y ∣ d − ε + ∣ x + y − z ∣ d − ε ) , where the notation dist Z d (( x, y ) , ( z, z )) is used to denote the euclidean distance in the lattice Z d betweenthe points ( x, y ) and ( z, z ) , the second inequality is obtained by computing the orthogonal projection of thepoint ( x, y ) ∈ Z d on the diagonal {( z, z ) ∈ Z d ∶ z ∈ Z d } and the third inequality is obtained by reducing thevalue of the constant c . Using the estimate (4.17), we deduce that ∑ z ∈ Z d ∣ z ∣ d − ε × (∣ x − z ∣ d − ε + ∣ y − z ∣ d − ε ) ≤ ∑ z ∈ Z d ∣ z ∣ d − ε × ∣ x − y ∣ d − ε + ∣ x + y − z ∣ d − ε (4.18) ≤ ∑ z ∈ Z d ∣ z ∣ d − ε × ∣ x − y ∣ d − ε + ∣ x + y − z ∣ d − ε ≤ C ∣ x − y ∣ d max (∣ x ∣ , ∣ y ∣) d − ε , where the computation in the third line is performed in Proposition 0.3 of Appendix C. Combining theestimates (4.16) and (4.18) completes the proof of the estimate (4.9).To estimate the term (4.8)-(ii), we use that 0 belongs to the support of the charge n q to write, for eachpair of points x , y ∈ supp n q ,(4.19) 1 ∣ x − x ∣ d − ε + ∣ y − y ∣ d − ε ≤ ∣ diam n q ∣ d ∣ x ∣ d − ε + ∣ y ∣ d − ε ≤ C q ∣ x ∣ d − ε + ∣ y ∣ d − ε . From the inequality (4.19), we deduce(4.20) (4.8) − ( ii ) = ∑ x ,y ∈ Z d ∣ n q ( x )∣ ∣ n q ( y )∣∣ x − x ∣ d − ε + ∣ y − y ∣ d − ε ≤ C q ∣ x ∣ d − ε + ∣ y ∣ d − ε . We then combine the estimates (4.8), (4.9), (4.20) to deduce the inequality, for each pair of points x, y ∈ Z d , ∥ u ( x, y, ⋅)∥ L ∞ ( µ β ) ≤ C q ∣ x − y ∣ d − ε max (∣ x ∣ , ∣ y ∣) d − ε + C q ∣ x ∣ d − ε + ∣ y ∣ d − ε (4.21) ≤ C q ∣ x − y ∣ d − ε max (∣ x ∣ , ∣ y ∣) d − ε . We then use this inequality to estimate the variance of the random variable U q ( x, ⋅) by using the formula (4.7).We obtainvar [U q ( z, ⋅)] ≤ C ∑ y,y ∈ Z d C q ∣ z − y ∣ d − ε max (∣ z ∣ , ∣ y ∣) d − ⋅ C ∣ y − y ∣ d − ⋅ C q ∣ z − y ∣ d − ε max (∣ z ∣ , ∣ y ∣) d − ε . We then use that the terms max (∣ z ∣ , ∣ y ∣) and max (∣ z ∣ , ∣ y ∣) are both larger than the value ∣ z ∣ to deduce thatvar [U q ( z, ⋅)] ≤ C q ∣ z ∣ d − ε ∑ y,y ∈ Z d ∣ z − y ∣ d − ε ⋅ ∣ y − y ∣ d − ⋅ ∣ z − y ∣ d − ε ≤ C q ∣ z ∣ d − ε . . HOMOGENIZATION OF THE MIXED DERIVATIVE OF THE GREEN’S MATRIX 141 The proof of the estimate (4.5) is complete.
We fix a radius R > m be the smallest integer such that the annulus A R is included in the cube ◻ m . The proof relies on atwo-scale expansion following the outline described in Section 2.2. We define the function H q by the formula(4.22) H q ∶= G q + ∑ i,j ∇ i G q ,j χ m,ij . We decompose the argument into three Steps.
Step 1.
In this step, we prove that the H − ( A R , µ β ) -norm of the term LH q is small, more specifically, weprove that there exists an exponent γ α > ∥LH q ∥ H − ( A R ,µ β ) ≤ C q R d − γ α . The proof is essentially identical to the argument presented in Section 3.2: we use the exact formula for thetwo-scale expansion H q given in (4.22) to compute the value of LH q and then use the quantitative propertiesof the corrector stated in Proposition 4.3 of Chapter 6 to prove that the H − ( A R , µ β ) -norm of the term LH q satisfies the estimate (4.23). Since the proof is rather long due to the technicalities caused by the specificstructure of the operator L (iterations of the Laplacian, sum over all the charges q ∈ Q ), we do not rewrite itbut only point out the main differences: ● We work in the annulus A R and not in the ball B R + δ , this difference makes the proof simpler sincewe do not have to take the additional parameter δ into considerations; ● We can always assume that the diameter of the charge q is smaller than R /
2, otherwise the constant C q is larger than R k for some large number k ∶= k ( d ) (since it is allowed to have an algebraic growthin the parameter ∥ q ∥ ) and the estimate (4.23) is trivial in this situation. Under the assumptiondiam q ≤ R /
2, we use the identity − a ∆ G q = A R instead of the identity − a ∆ G δ = ρ δ in the ball B R + δ ; ● We use the regularity estimates on the function G q stated in Proposition 4.1 instead of the estimateson the Green’s function G stated in Proposition 3.1. Since the map G q scales like the gradient ofthe Green’s function (in particular it decays like ∣ x ∣ − d ), we obtain an additional factor R in the rightside of (4.23) compared to (3.7), i.e., we obtain ∥LH q ∥ H − ( A R ,µ β ) ≤ C q R d − γ α instead of ∥LH δ, ⋅ k − ρ δ, ⋅ k ∥ H − ( A R ,µ β ) ≤ CR d − − γ α . Step 2.
In this step, we use the main result (4.23) of Substep 3.1 to prove that the gradient of the Green’sfunction ∇U q is close to the gradient of the two-scale expansion ∇H q in the L ( A R , µ β ) -norm. We prove theestimate(4.24) ∥∇U q − ∇H q ∥ L ( A R ,µ β ) ≤ C q R d + γ α . To simplify the rest of the argument, we do not prove the estimate (4.24) directly. We slightly reduce thesize of the annulus A R and define the set A R to be the annulus A R ∶= { x ∈ Z d ∶ . R ≤ ∣ x ∣ ≤ . R } . We notethat we have the inclusion, for each radius R ≥ A R ⊆ A R . In this substep, we prove the inequality(4.25) ∥∇U q − ∇H q ∥ L ( A R ,µ β ) ≤ C q R d + γ δ . The inequality (4.24) can then be deduced from (4.25) by a covering argument.The argument is similar to the one presented in Section 3.3 except that, instead of making use of themollifier exponent δ to prove that the H -norm is of the difference (∇H δ − G δ ) is small, as it was done in theestimates (3.66) and (3.67), we use the main result (4.6) of Section 4.3. We first let η be a cutoff functionwhich satisfies the properties:(4.26) 0 ≤ η ≤ , supp η ⊆ A R , η = A R , ∀ k ∈ N , ∣∇ k η ∣ ≤ CR k .
42 7. QUANTITATIVE HOMOGENIZATION OF THE GREEN’S MATRIX
We then use the function η (U q − H q ) as a test function in the definition of the H − ( A R , µ β ) of the inequal-ity (4.23) and use the identity LU q = A R × Ω. We obtain1 R d ∑ x ∈ A R ⟨ η (U q − H q ) L (U q − H q )⟩ µ β ≤ ∥L (U q − H q )∥ H − ( B R + δ ,µ β ) ∥ η (U q − H q )∥ H ( A R ,µ β ) (4.27) ≤ CR d + γ α ∥ η (U q − H q )∥ H ( A R ,µ β ) . We then estimate the H ( A R , µ β ) -norm of the function U q − H q with similar arguments as the one presentedin the proof of the inequality (3.52), the only difference is that we use the regularity estimates stated inProposition 4.1 instead of the regularity estimates for the functions G δ and H . We obtain(4.28) ∥ η (U q − H q )∥ H ( A R ,µ β ) ≤ ∥ η U q ∥ H ( A R ,µ β ) + ∥ η H q ∥ H ( A R ,µ β ) ≤ C q R d − − ε . For later use, we also note that the same argument yields to the inequality(4.29) ∥∇U q − ∇H q ∥ L ( A R ,µ β ) ≤ ∥∇U q ∥ L ( A R ,µ β ) + ∥∇H q ∥ L ( A R ,µ β ) ≤ C q R d − ε . We then combine the inequalities (4.25) and (4.27) and use that ε ≪ γ α to deduce that(4.30) 1 R d ∑ x ∈ A R ⟨ η (U q − H q ) L (U q − H q )⟩ µ β ≤ C q R d + γ α . Thus to prove the inequality (4.25), it is sufficient to prove the estimate ∥∇U q − ∇H q ∥ L ( A R ,µ β ) ≤ R d ∑ x ∈ A R ⟨ η (U q − H q ) L (U q − H q )⟩ µ β + C q R d + γ δ . First, by definition of the Helffer-Sj¨ostrand operator L , we have the identity ∑ x ∈ Z d ⟨ η (U q − H q ) L (U q − H q )⟩ µ β = ∑ x,y ∈ Z d η ( x ) ⟨( ∂ y U q ( x, ⋅) − ∂ y H q ( x, ⋅)) ⟩ µ β (4.31) + β ∑ x ∈ Z d ⟨(∇U q − ∇H q ) ( x, ⋅) ⋅ ∇ ( η (U q − H q )) ( x, ⋅)⟩ µ β + ∑ q ∈Q ⟨∇ q (U q − H q ) ⋅ a q ∇ q ( η (U q − H q ))⟩ µ β + β ∑ n ≥ ∑ x ∈ Z d β n ⟨∇ n + (U q − H q ) ( x, ⋅) ⋅ ∇ n + ( η (U q − H q )) ( x, ⋅)⟩ µ β . We then estimate the four terms on the right side separately. For the first one, we use that it is non-negative(4.32) ∑ x,y ∈ Z d η ( x ) ⟨( ∂ y U q ( x, ⋅) − ∂ y H q ( x, ⋅)) ⟩ µ β ≥ . For the second one, we expand the gradient of the product η (U q − H q ) and write ∑ x ∈ Z d ⟨(∇U q − ∇H q ) ( x, ⋅) ⋅ ∇ ( η (U q − H q )) ( x, ⋅)⟩ µ β (4.33) = ∑ x ∈ Z d η ( x ) ⟨(∇U q − ∇H q ) ( x, ⋅) ⋅ ∇ (U q − H q ) ( x, ⋅)⟩ µ β + ∑ x ∈ Z d ⟨(∇U q − ∇H q ) ( x, ⋅) ⋅ ∇ η ( x ) (U q − H q ) ( x, ⋅)⟩ µ β . Dividing the identity (4.33) by the volume factor R d and using the properties of the function η stated in (4.26),we obtain(4.34) R − d ∑ x ∈ Z d ⟨(∇U q ( x, ⋅) − ∇H q ) ( x, ⋅) ⋅ ∇ ( η (U q − H)) ( x, ⋅)⟩ µ β ≥ c ∥ η (∇U q − ∇H q )∥ L ( A R ,µ β ) − CR ∥∇U q − ∇H q ∥ L ( A R ,µ β ) ∥U q − H q ∥ L ( A R ,µ β ) . . HOMOGENIZATION OF THE MIXED DERIVATIVE OF THE GREEN’S MATRIX 143 We then use the inequality (4.6) and the estimate (4.29) and the quantitative sublinearity of the corrector todeduce that(4.35) 1 R ∥∇U q − ∇H q ∥ L ( A R ,µ β ) ∥U q − H q ∥ L ( A R ,µ β ) ≤ R ⋅ CR d − ε ⋅ CR d − + γ δ ≤ CR d + γ δ . We then combine the inequalities (4.34) and (4.35) to deduce that(4.36) R − d ∑ x ∈ Z d ⟨(∇U q ( x, ⋅) − ∇H q ) ( x, ⋅) ⋅ ∇ ( η (U q − ∇H q )) ( x, ⋅)⟩ µ β + C q R d + γ δ ≥ c ∥ η (∇U q − ∇H q )∥ L ( A R ,µ β ) . The two remaining terms in the right side of the estimate (4.31) (involving the iteration of the Laplacian andthe sum over the charges) are estimated following the ideas developed in in Section 3.3 (see (3.70) and (3.71))or in the proof of the Caccioppoli inequality (Proposition 1.1 of Chapter 5). We skip the details and write theresult: we obtain(4.37) R − d ∑ q ∈Q ⟨∇ q (U q − ∇H q ) ⋅ a q ∇ q ( η (U q − ∇H q ))⟩ µ β + C q R ( d − γ δ ) ≥ − Ce − c √ β ∥ η (∇U q − ∇H q )∥ L ( A R ,µ β ) and(4.38) R − d ∑ n ≥ ∑ x ∈ Z d β n ⟨∇ n + (U q − H q ) ( x, ⋅) ⋅ ∇ n + ( η (U q − H q )) ( x, ⋅)⟩ µ β + C q R ( d + γ δ ) ≥ . We then combine the estimates (4.32), (4.36), (4.37) and (4.38) with the identity (4.31), choose the inversetemperature β large enough so that the right side of (4.37) can be absorbed by the right side of (4.34) anduse that the cutoff function η is equal to 1 in the annulus A R . We obtain(4.39) ∥∇U q − ∇H q ∥ L ( A R ,µ β ) ≤ CR d ∑ x ∈ Z d ⟨ η (U q − H q ) L (U q − H q )⟩ µ β + C q R d + γ δ . We then combine the inequality (4.39) with the estimate (4.30) to complete the proof of (4.25). Step 2 iscomplete.
Step 3. The conclusion.
In this step, we prove the L -estimate(4.40) XXXXXXXXXXX∇U q − ∑ i,j ( e ij + ∇ χ ij ) ∇ G q ,j XXXXXXXXXXX L ( A R ,µ β ) ≤ C q R d + γ δ . In view of the estimate (4.25) proved in Step 2, it is sufficient to prove the inequality(4.41)
XXXXXXXXXXX∇H q − ∑ i,j ( e ij + ∇ χ ij ) ∇ G q ,j XXXXXXXXXXX L ( A R ,µ β ) ≤ C q R d + γ δ . The proof of (4.41) relies on the regularity estimate on the function G q stated in Proposition 4.1, thequantitative sublinearity of the corrector stated in Proposition 4.3 of Chapter 6 and the quantitative estimatefor the difference of the finite and infinite-volume gradient of the corrector stated in Proposition 4.4 of Chapter 6.The argument is identical (and even simpler since we do not have to take into account the parameter δ ) to theargument given in Section 3.4 so we skip the details. The proof of Step 3, and thus of Theorem 4, is complete.HAPTER 8 Proof of the estimates in Chapter 4
In this chapter, we present the proofs of the technical lemmas which are used in Chapter 4 to proveTheorem 1. Most of the heuristic of the arguments are presented in this chapter and we refer to it for anoverview of the results. As it may be useful to the reader, we record below the tools established in this articlewhich are used in this chapter: ● In Sections 1, 2 and 3, we study the correlation of random variables; this is achieved by using theHelffer-Sj¨ostrand representation formula. We need to use the properties of the Green’s functionassociated to the Helffer-Sj¨ostrand operator stated in Proposition 4.7 of Chapter 3; ● In Section 3, we need to study the correlation between a solution of a Helffer-Sj¨ostrand equation andthe random variables X x and Y . To this end, we appeal Helffer-Sj¨ostrand representation formulaand the differentiated Helffer-Sj¨ostrand equation as well as to the properties of the Green’s functionassociated to this operator stated in Proposition 4.2 of Chapter 5; ● Sections 4 and 5 are devoted to the proofs of some properties of the discrete Green’s function on thelattice Z d ; they can be read independently of the rest of the article.
1. Removing the terms X sin cos , X cos cos and X sin sin We recall the definitions of the values Z β ( σ ) and Z β ( ) introduced in (1.9) of Chapter 3, the definitionsof the random variables Y , X x , X sin cos , X cos cos , X sin sin introduced in (1.4) of Chapter 4 and the identity(1.1) Z β ( σ ) Z β ( ) = ⟨ Y X x X sin cos X cos cos X sin sin ⟩ µ β . Given a charge q ∈ Q , we recall the conventional notation C q to mean the the constant C depends on thevariables d, β and q and that the dependence in the q variable is at most algebraic, i.e., there exists an exponent k ∶= k ( d ) and a constant C ∶= C ( d, β ) such that C q ≤ C ∥ q ∥ k . We also recall the notation, for each point y ∈ Z d , Q y ∶= { q ∈ Q ∶ y ∈ supp n q } . Lemma . There exist constants c ∶= c ( d, β ) and C ∶= C ( d, β ) < ∞ such that (1.2) Z β ( σ ) Z β ( ) = ⟨ Y X x ⟩ µ β + c ⟨ Y X x ⟩ µ β ∣ x ∣ d − + O ( C ∣ x ∣ d − ) . As a consequence, the following statements are equivalent (1.3) ∃ γ ∈ ( , ∞) , ∃ c , c ∈ R , Z β ( σ ) Z β ( ) = c + c ∣ x ∣ d − + O ( C ∣ x ∣ d − + γ )⇐⇒ ∃ γ ∈ ( , ∞) , ∃ c , c ∈ R , ⟨ Y X x ⟩ µ β = c + c ∣ x ∣ d − + O ( C ∣ x ∣ d − + γ ) . Remark . The values of the constants c , c in the two sides of the equivalence (1.3) are not necessarilyequal; they are related through the constant c which appears in (1.2). We use the same notation because weare not interested in their specific values but only in their existence. Proof.
As is explained in Section 2 of Chapter 4, the proof of estimate (1.2) is based on the proof of thefollowing estimates(1.4) ⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩ ∥ X sin cos − ∥ L ∞ ≤ C ∣ x ∣ d − , ∥ X cos cos − ∥ L ∞ ≤ C ∣ x ∣ d − , var µ β X sin sin ≤ C ∣ x ∣ d − , E [ X sin sin ] = + c ∣ x ∣ d − + O ( C ∣ x ∣ d − ) . We first prove that (1.4) implies (1.2). From the identity (1.1) and a direct computation involving theCauchy-Schwarz inequality, we obtain the estimate ∣⟨ Y X x X sin cos X cos cos X sin sin ⟩ µ β − ⟨ Y X x ⟩ µ β ⟨ X sin sin ⟩ µ β ∣ ≤ ∥ Y X x ∥ L ( µ β ) ( var µ β X sin sin ) + ⟨ Y X x X sin sin ⟩ µ β (∥ X sin cos − ∥ L ∞ + ∥ X cos cos − ∥ L ∞ ) . Using the Brascamp-Lieb inequality to estimate the L ( µ β ) -norm of the random variables Y X x and theestimates on the random variable X sin sin stated in (1.4), we write(1.5) ∥ Y X x ∥ L ( µ β ) ≤ ∥ Y ∥ L ( µ β ) ∥ X x ∥ L ( µ β ) ≤ C and ⟨ Y X x X sin sin ⟩ µ β ≤ ∥ Y X x ∥ L ( µ β ) ∥ X sin sin ∥ L ( µ β ) ≤ C ∥ Y X x ∥ L ( µ β ) ( E [ X sin sin ] + var X sin sin ) ≤ C. We combine these inequalities with the estimates (1.4) to deduce the the estimate ∣⟨ Y X x X sin cos X cos cos X sin sin ⟩ µ β − ⟨ Y X x ⟩ µ β ( + c ∣ x ∣ d − + O ( C ∣ x ∣ d − ))∣ ≤ C ∣ x ∣ d − . The expansion (1.2) is then a direct consequence of the identity (1.1), the estimate (1.4) and the upperbound (1.5).It remains to prove the estimates stated in (1.4); we first focus on the first two inequalities involving therandom variables X sin cos and X cos cos . The proof relies on the following ingredients: ● For each point y ∈ Z d and each charge q ∈ Q y , we have the estimate (∇ G, n q ) ≤ ∥∇ G ∥ L ∞ ( supp n q ) ∥ n q ∥ L ≤ sup z ∈ supp n q C ∣ z ∣ d − ∥ n q ∥ L ≤ sup ∣ z ∣≤ diam n q C ∣ y + z ∣ d − ∥ n q ∥ L ≤ C ( diam n q ) d − ∣ y ∣ d − ∥ n q ∥ L ≤ C q ∣ y ∣ d − , where we used in the second inequality that, for each charge q in the set Q y , the support of n q isincluded in the ball B ( y, diam n q ) . A similar computation shows the estimate (∇ G x , n q ) ≤ C q ∣ y − x ∣ d − ; ● The standard estimates, for each real number a ∈ R , ∣ sin a ∣ ≤ ∣ a ∣ , ∣ cos a − ∣ ≤ ∣ a ∣ and the estimate,for each charge q ∈ Q , ∣ z ( β, q )∣ ≤ e − c √ β ∥ q ∥ . . REMOVING THE TERMS X sincos , X coscos AND X sinsin We obtain the inequality
RRRRRRRRRRR ∑ q ∈Q z ( β, q ) sin 2 π ( φ, q ) sin 2 π (∇ G x , n q ) ( cos 2 π (∇ G, n q ) − )RRRRRRRRRRR ≤ C ∑ y ∈ Z d ∑ q ∈Q y e − c √ β ∥ q ∥ C q ∣ y − x ∣ d − ∣ y ∣ d − (1.6) ≤ C ∑ y ∈ Z d ∣ y − x ∣ d − ∣ y ∣ d − ≤ C ∣ x ∣ d − , where we used the exponential decay of the term e − c √ β ∥ q ∥ to absorb the algebraic growth of the constant C q .With a similar strategy, we obtain the two inequalities(1.7) RRRRRRRRRRR ∑ q ∈Q z ( β, q ) sin 2 π ( φ, q ) sin 2 π (∇ G, n q ) ( cos 2 π (∇ G x , q ) − )RRRRRRRRRRR ≤ C ∣ x ∣ d − , RRRRRRRRRRR ∑ q ∈Q z ( β, q ) sin 2 π ( φ, q ) ( cos (∇ G x , q ) − ) ( cos (∇ G, q ) − )RRRRRRRRRRR ≤ C ∣ x ∣ d − . We then combine the estimates (1.6) and (1.7) and use that the exponential function is Lipschitz on anycompact subset of R to obtain, for each realization of the field φ ∈ Ω, ∣ X sin cos ( φ ) − ∣ ≤ C ∣ x ∣ d − and ∣ X cos cos ( φ ) − ∣ ≤ C ∣ x ∣ d − . This result implies the L ∞ ( µ β ) -estimates stated in (1.4).There remains to prove the estimates corresponding to the variance and the expectation of the randomvariable X sin sin in (1.4). We first note that a computation similar to the one performed in (1.6) gives thefollowing L ∞ ( µ β ) -estimate: for each realization of the field φ ∈ Ω,(1.8)
RRRRRRRRRRR ∑ q ∈Q z ( β, q ) cos 2 π ( φ, q ) sin (∇ G, n q ) sin 2 π (∇ G x , n q )RRRRRRRRRRR ≤ C ∑ y ∈ Z d ∣ y − x ∣ d − ∣ y ∣ d − ≤ C ∣ x ∣ d − . By the estimate (1.8) and the Taylor expansion of the exponential, we obtain the bound
RRRRRRRRRRR X sin sin − − ∑ q ∈Q z ( β, q ) cos 2 π ( φ, q ) sin 2 π (∇ G, n q ) sin 2 π (∇ G x , n q )RRRRRRRRRRR≤ C ⎛⎝ ∑ q ∈Q z ( β, q ) cos 2 π ( φ, q ) sin 2 π (∇ G, n q ) sin 2 π (∇ G x , n q )⎞⎠ ≤ C ∣ x ∣ d − . Since the dimension d is assumed to be larger than 3, we have the inequality 2 d − ≥ d −
1. We deduce that toprove the estimates pertaining to the random variable X sin sin in (1.4), it is sufficient to prove the inequalities(1.9) var ⎡⎢⎢⎢⎢⎣ ∑ q ∈Q z ( β, q ) cos 2 π ( φ, q ) sin 2 π (∇ G, n q ) sin 2 π (∇ G x , n q )⎤⎥⎥⎥⎥⎦ ≤ C ∣ x ∣ d − and the expansion(1.10) E ⎡⎢⎢⎢⎢⎣ ∑ q ∈Q z ( β, q ) cos 2 π ( φ, q ) sin 2 π (∇ G, n q ) sin 2 π (∇ G x , n q )⎤⎥⎥⎥⎥⎦ = c ∣ x ∣ d − + O ( C ∣ x ∣ d − ) . The estimate (1.9) involving the variance can be estimated by the Helffer-Sj¨ostrand representation formulaand the bounds on the Green’s matrix G stated in Proposition 4.7 of Chapter 3. We first note that, for each
48 8. PROOF OF THE ESTIMATES IN CHAPTER 4 point y ∈ Z d ,(1.11) ∂ y ⎛⎝ ∑ q ∈Q z ( β, q ) cos 2 π ( φ, q ) sin 2 π (∇ G, n q ) sin 2 π (∇ G x , n q )⎞⎠= − ∑ q ∈Q πz ( β, q ) sin 2 π ( φ, q ) sin 2 π (∇ G, q ) sin 2 π (∇ G x , q ) q ( y ) . From the identity (1.11), we deduce that to compute the variance (1.9), one needs to solve the Helffer-Sj¨ostrandequation
LW( y, φ ) = − ∑ q ∈Q z ( β, q ) sin 2 π ( φ, q ) sin 2 π (∇ G, q ) sin 2 π (∇ G x , q ) q ( y ) . Using the notation G for the Green’s function associated to the Helffer-Sj¨ostrand operator L introduced inSection 3.3 of Chapter 3, we have the identity(1.12) W( y, φ ) = − ∑ q ∈Q z ( β, q ) sin 2 π (∇ G, n q ) sin 2 π (∇ G x , n q ) ∑ z ∈ supp n q d ∗ z G sin 2 π ( φ,q ) ( y, φ ; z ) n q ( z ) . Taking the exterior derivative of the identity (1.12) shows the equalityd ∗ W( y, φ ) = − ∑ q ∈Q z ( β, q ) sin 2 π (∇ G, n q ) sin 2 π (∇ G x , n q ) ∑ z ∈ supp n q d ∗ y d ∗ z G sin 2 π ( φ,q ) ( y, φ ; z ) n q ( z ) . Using the estimate on the Green’s function proved in Proposition 4.7 of Chapter 3, and the fact that thecodifferential d ∗ is a linear functional of the gradient, we deduce the estimate, for each pair of points y, z ∈ Z d ,(1.13) ∥ d ∗ y d ∗ z G sin 2 π ( φ,q ) ( y, φ ; z )∥ L ∞ ( µ β ) ≤ C ∣ y − z ∣ d − ε . Using the estimate (1.13) and a computation similar to the one performed in (1.6), we obtain the inequality,for each point y ∈ Z d , ∥ d ∗ W( y, ⋅)∥ L ∞ ( µ β ) ≤ ∑ z ∈ Z d ∑ q ∈Q z e − c √ β ∥ q ∥ C q ∣ z ∣ d − ∣ z − x ∣ d − ∣ y − z ∣ d − ε (1.14) ≤ ∑ z ∈ Z d C ∣ z ∣ d − ∣ z − x ∣ d − ∣ y − z ∣ d − ε . Using the definition of the map W , we apply the Helffer-Sj¨ostrand representation formula and deduce thatvar ⎡⎢⎢⎢⎢⎣ ∑ q ∈Q z ( β, q ) cos 2 π ( φ, q ) sin 2 π (∇ G, n q ) sin 2 π (∇ G x , n q )⎤⎥⎥⎥⎥⎦≤ π ∑ y ∈ Z d ⟨⎛⎝ ∑ q ∈Q z ( β, q ) sin 2 π ( φ, q ) sin 2 π (∇ G, n q ) sin 2 π (∇ G x , q ) n q ( y )⎞⎠ d ∗ W( y, φ )⟩ µ β . Using the estimates (1.14) and a computation similar to the one performed in (1.6), we deduce thatvar ⎡⎢⎢⎢⎢⎣ ∑ q ∈Q z ( β, q ) cos 2 π ( φ, q ) sin 2 π (∇ G, n q ) sin 2 π (∇ G x , n q )⎤⎥⎥⎥⎥⎦ (1.15) ≤ ∑ y ∈ Z d ∑ q ∈Q y e − c √ β ∥ q ∥ C q ∣ y ∣ d − ∣ x − y ∣ d − ∥ d ∗ W( y, ⋅)∥ L ∞ ( µ β ) ≤ C ∑ y,z ∈ Z d ∣ y ∣ d − ∣ x − y ∣ d − × ∣ z ∣ d − ∣ z − x ∣ d − × ∣ y − z ∣ d − ε ≤ C ∣ x ∣ d − , where we used Proposition 0.3 of Appendix C in the last line. There only remains to prove the identity (1.10).To this end, we use the ideas presented in Section 5.2 of Chapter 4. We first define an equivalence relation onthe set Q : one says that two charges q and q ′ are equivalent, and denote it by q ∼ q ′ , if they are equal up to atranslation, i.e., q ∼ q ′ ⇐⇒ ∃ y ∈ Z d such that q (⋅ + y ) = q ′ . . REMOVING THE TERMS X sincos , X coscos AND X sinsin We denote this quotient space by Q/ Z d and for each charge q ∈ Q , we denote by [ q ] its equivalence class. Foreach equivalence class [ q ] ∈ Q/ Z d , we select a charge q ∈ Q such that 0 belongs to the support of n q (if there ismore than one candidate, we break ties by using an arbitrary criterion). We note that, for each charge q ∈ Q ,by the definition of the charge n q and the coefficient z ( β, q ) , we have the identities, for each point z ∈ Z d ,(1.16) z ( β, q ) = z ( β, q (⋅ − z )) and n q (⋅− z ) = n q (⋅ − z ) . Additionally, we can decompose the sum over the charges q ∈ Q along the equivalence classes, i.e., we can write,for any non-negative or summable (with respect to the counting measure on the set Q ) function F ∶ Q → R (1.17) ∑ q ∈Q F ( q ) = ∑ [ q ]∈Q/ Z d ∑ z ∈ Z d F ( q (⋅ − z )) . We can thus decompose the sum ∑ q ∈Q z ( β, q ) cos 2 π ( φ, q ) sin 2 π (∇ G, n q ) sin 2 π (∇ G x , n q )= ∑ [ q ]∈Q/ Z d z ( β, q ) ∑ y ∈ Z d cos 2 π ( φ, q ( y + ⋅)) sin 2 π (∇ G, n q ( y + ⋅)) sin 2 π (∇ G x , n q ( y + ⋅)) . Taking the expectation, using the translation invariance of the measure µ β and using the identities (∇ G, n q ( y +⋅)) = (∇ G (⋅ − y ) , n q ) and (∇ G x , n q ( y + ⋅)) = (∇ G (⋅ − y ) , n q ) , we deduce that(1.18) E ⎡⎢⎢⎢⎢⎣ ∑ q ∈Q z ( β, q ) cos 2 π ( φ, q ) sin 2 π (∇ G, n q ) sin 2 π (∇ G x , n q )⎤⎥⎥⎥⎥⎦= ∑ [ q ]∈Q/ Z d z ( β, q ) E [ cos 2 π ( φ, q )] ∑ y ∈ Z d sin 2 π (∇ G (⋅ − y ) , n q ) sin 2 π (∇ G x (⋅ − y ) , n q ) . Fix an equivalence class [ q ] ∈ Q/ Z d and define the value ( n q ) ∶= ∑ z ∈ Z d n q ( z ) ∈ R d (which only depends on theequivalence class of the charge q ). We prove the expansion(1.19) ∑ y ∈ Z d sin 2 π (∇ G (⋅ − y ) , n q ) sin 2 π (∇ G x (⋅ − y ) , n q ) = π ∑ y ∈ Z d ∇ G ( y ) ⋅ ( n q ) × ∇ G x ( y ) ⋅ ( n q ) + O ( C q ∣ x ∣ d − ) . With the same arguments as the ones presented in (5.21) of Chapter 4, we write the inequalities ∣(∇ G (⋅ − y ) , n q ) − ∇ G ( y ) ⋅ ( n q )∣ ≤ C q ∣ y ∣ d and ∣(∇ G x (⋅ − y ) , n q ) − ∇ G x ( y ) ⋅ ( n q )∣ ≤ C q ∣ y + x ∣ d . Combining this result with the estimate, for each real number a ∈ R , ∣ sin a − a ∣ ≤ ∣ a ∣ , we obtain the estimate ∣ sin 2 π (∇ G (⋅ − y ) , n q ) sin 2 π (∇ G x (⋅ − y ) , n q ) − π ∇ G ( y ) ⋅ ( n q ) × ∇ G ( y + x ) ⋅ ( n q )∣≤ C q ∣ y ∣ d ∣ x − y ∣ d − + C q ∣ y ∣ d − ∣ x − y ∣ d . Summing over the points y ∈ Z d completes the proof of the identity (1.19). It remains to prove that theexpansion (1.19) implies the estimate (1.10). To this end, we first note that, if we denote by ( n q ) , . . . , ( n q ) d the d -coordinates of the vector ( n q ) , then we have the equality ∑ y ∈ Z d ∇ G ( y ) ⋅ ( n q ) × ∇ G x ( y ) ⋅ ( n q ) = d ∑ i,j = ( n q ) i ( n q ) j ∑ y ∈ Z d ∇ i G ( y )∇ j G x ( y ) . We sum over all the equivalence class [ q ] ∈ Q/ Z d and use the exponential decay of the term z ( β, q ) to absorbthe (at most) algebraic growth of the various terms involving the charge q . We obtain(1.20) ∑ [ q ]∈Q/ Z d z ( β, q ) E [ cos 2 π ( φ, q )] ∑ y ∈ Z d sin 2 π (∇ G (⋅ − y ) , n q ) sin 2 π (∇ G x (⋅ − y ) , n q )= d ∑ i,j = c ij ∑ y ∈ Z d ∇ i G ( y )∇ j G x ( y ) + O ( C ∣ x ∣ d − ) ,
50 8. PROOF OF THE ESTIMATES IN CHAPTER 4 where the constants are defined by the formulas c ij = π ∑ [ q ]∈Q/ Z d z ( β, q ) E [ cos 2 π ( φ, q )] ( n q ) i ( n q ) j . By combining the estimates (1.18) and (1.20), we have obtained the expansion(1.21) E ⎡⎢⎢⎢⎢⎣ ∑ q ∈Q z ( β, q ) cos 2 π ( φ, q ) sin 2 π (∇ G, n q ) sin 2 π (∇ G x , n q )⎤⎥⎥⎥⎥⎦ = d ∑ i,j = c ij ∑ y ∈ Z d ∇ i G ( y )∇ j G x ( y ) + O ( C ∣ x ∣ d − ) . The expansion (1.21) is not exactly (1.10). To complete the argument, we appeal to the symmetry in-variance of the dual Villain model following the argument presented in Section 5.1 of Chapter 4 andthe results proved in Section 4 of this chapter. We let H be the group of lattice-preserving maps in-troduced in Chapter 2. Since the measure µ β is invariant under the elements of the group H , the map x ↦ E [∑ q ∈Q z ( β, q ) cos 2 π ( φ, q ) sin 2 π (∇ G, n q ) sin 2 π (∇ G x , n q )] satisfies the same invariance property. We canthus apply Proposition 4.4 of Section 4 (whose proof is independent of the rest of the arguments developed inthe article) to complete the proof of (1.10). (cid:3)
2. Removing the contributions of the cosines
The goal of this section is to prove Lemma 2.1 of Chapter 4, which is restated below. We recall thedefinitions of the charges Q x and n Q x stated in (3.5) and (3.8) of Chapter 4: for each pair ( y, φ ) ∈ Z d × Ω,(2.1) Q x ( y, φ ) ∶= ∑ q ∈Q πz ( β, q ) cos 2 π ( φ, q ) sin 2 π (∇ G x , n q ) q ( y ) and(2.2) n Q x ( y, φ ) ∶= ∑ q ∈Q πz ( β, q ) cos 2 π ( φ, q ) sin 2 π (∇ G x , n q ) n q ( y ) . The statement of Lemma 2.1 is recalled below.
Lemma . One has the identity (2.3) cov [ X x , Y ] = ∑ y ∈ Z d ⟨ X x Q x ( y, ⋅)V( y, ⋅)⟩ µ β + O ( C ∣ x ∣ d − − ε ) , where V ∶ Z d × Ω → R ( d ) is the solution of the Helffer-Sj¨ostrand equation, for each ( y, φ ) ∈ Z d × Ω , (2.4) LV( y, φ ) = Q ( y ) Y ( φ ) . Proof.
We start from the Helffer-Sj¨ostrand representation formula stated in (3.3) of Chapter 4 andrecalled below(2.5) cov [ X x , Y ] = ∑ y ∈ Z d ⟨( ∂ y X x ) Y( y, ⋅)⟩ µ β , where Y ∶ Z d × Ω → R ( d ) is the solution of the Helffer-Sj¨ostrand equation, for each ( y, φ ) ∈ Z d × Ω,(2.6)
LY( y, φ ) = ∂ y Y ( φ ) . Using the definition of the random variables Y and X x stated in (1.4) of Chapter 4, we have the identities, foreach y ∈ Z d ,(2.7) ∂ y Y ( φ ) = − ⎛⎝ Q ( y, φ ) +
12 2 π ∑ q ∈Q z ( β, q ) sin 2 π ( φ, q ) ( cos 2 π (∇ G, n q ) − ) q ( y )⎞⎠ Y ( φ ) and(2.8) ∂ y X x ( φ ) = − ⎛⎝ Q x ( y, φ ) + ∑ q ∈Q
12 2 πz ( β, q ) sin 2 π ( φ, q ) ( cos 2 π (∇ G x , n q ) − ) q ( y )⎞⎠ X x ( φ ) . . REMOVING THE CONTRIBUTIONS OF THE COSINES 151 The objective of the proof is to remove the terms involving the cosine in the right side of the identities (2.7)and (2.8). The proof requires to use the following estimates established in (3.9) and (3.11) of Chapter 4: foreach point y ∈ Z d ,(2.9) ∥ n Q x ( y, ⋅)∥ L ∞ ( µ β ) ≤ C ∣ y − x ∣ d − and(2.10) RRRRRRRRRRR ∑ q ∈Q z ( β, q ) sin 2 π ( φ, q ) ( cos 2 π (∇ G x , n q ) − ) n q ( y )RRRRRRRRRRR ≤ C ∣ y − x ∣ d − . The same arguments give the estimates(2.11) ∥ n Q ( y, ⋅)∥ L ∞ ( µ β ) ≤ C ∣ y ∣ d − , and(2.12) RRRRRRRRRRR ∑ q ∈Q z ( β, q ) sin 2 π ( φ, q ) ( cos 2 π (∇ G , n q ) − ) n q ( y )RRRRRRRRRRR ≤ C ∣ y ∣ d − . We split the argument into three steps: ● In Step 1, we prove that the solution of the Helffer-Sj¨ostrand equation Y satisfies the upper bound,for each y ∈ Z d ,(2.13) ∥ d ∗ Y( y, ⋅)∥ L ( µ β ) ≤ C ∣ y ∣ d − − ε ; ● In Step 2, we prove that the covariance between the random variables X x and Y satisfies theexpansion(2.14) cov [ X x , Y ] = ∑ y ∈ Z d ⟨ X x Q x ( y, ⋅)Y( y, ⋅)⟩ µ β + O ( C ∣ x ∣ d − − ε ) ; ● In Step 3, we use the symmetry of the Helffer-Sj¨ostrand operator L to complete the proof ofLemma 2.1. Step 1.
We first express the function Y in terms of the Green function associated to the Helffer-Sj¨ostrandoperator L . From the equation (3.4) of Chapter 4, we deduce the formula, for each ( y, φ ) ∈ Z d × Ω,(2.15) Y( y, φ ) = ∑ y ∈ Z d ∑ q ∈Q πz ( β, q )[ sin 2 π (∇ G, n q ) d ∗ y G cos 2 π (⋅ ,q ) Y ( y, φ ; y ) n q ( y )+ ( cos 2 π (∇ G, n q ) − ) n q ( y ) d ∗ y G sin 2 π (⋅ ,q ) Y ( y, φ ; y ) n q ( y )] . From the identity (2.15), we deduce the following formula for the function d ∗ Y d ∗ Y( y, φ ) = π ∑ y ∈ Z d ∑ q ∈Q z ( β, q ) sin 2 π (∇ G, n q ) d ∗ y d ∗ y G cos 2 π (⋅ ,q ) Y ( y, φ ; y ) n q ( y )+ π ∑ y ∈ Z d ∑ q ∈Q ( cos 2 π (∇ G, n q ) − ) d ∗ y d ∗ y G sin 2 π (⋅ ,q ) Y ( y, φ ; y ) n q ( y ) . Using the estimate on the Green’s function proved in Proposition 4.7 of Chapter 3, that the random variable Y belongs to the space L ( µ β ) and that the codifferential d ∗ is a linear functional of the gradient, we obtainthe estimate, for each pair of points y, y ∈ Z d , ∥ d ∗ y d ∗ y G cos 2 π (⋅ ,q ) Y ( y, φ ; y )∥ L ( µ β ) ≤ C ∥ cos 2 π ( φ, q ) Y ∥ L ( µ β ) ∣ y − y ∣ d − ε ≤ C ∥ Y ∥ L ( µ β ) ∣ y − y ∣ d − ε ≤ C ∣ y − y ∣ d − ε and, with the same argument, ∥ d ∗ y d ∗ y G sin 2 π (⋅ ,q ) Y ( y, φ ; y )∥ L ( µ β ) ≤ C ∣ y − y ∣ d − ε .
52 8. PROOF OF THE ESTIMATES IN CHAPTER 4
We then combine the two previous inequalities with the estimates (2.11) and (2.12). We obtain the inequality ∥ d ∗ Y( y, φ )∥ L ( µ β ) ≤ ∑ y ∈ Z d C ∣ y ∣ d − ∥ d ∗ y d ∗ y G cos 2 π (⋅ ,q ) Y ( y, φ ; y )∥ L ( µ β ) + ∑ y ∈ Z d C ∣ y ∣ d − ∥ d ∗ y d ∗ y G sin 2 π (⋅ ,q ) Y ( y, φ ; y )∥ L ( µ β ) ≤ ∑ y ∈ Z d C ∣ y ∣ d − ∣ y − y ∣ d − ε + C ∣ y − y ∣ d − ∣ y − y ∣ d − ε ≤ ∑ y ∈ Z d C ∣ y ∣ d − − ε . The proof of Step 1 is complete.
Step 2.
By the Helffer-Sj¨ostrand formula (2.5), we have the identitycov [ X x , Y ] = ∑ y ∈ Z d ⟨( ∂ y X x ) Y( y, ⋅)⟩ µ β = ∑ y ∈ Z d ⟨ Q x ( y ) X x Y( y, ⋅)⟩ µ β −
12 2 π ⟨ ∑ q ∈Q z ( β, q ) sin 2 π ( φ, q ) ( cos 2 π (∇ G x , n q ) − ) n q ( y ) X x d ∗ Y( y, ⋅)⟩ µ β . (2.16)The objective of this step is to prove that the term involving the cosine in the right side of (2.16) is of lowerorder; specifically we prove the inequality, for each y ∈ Z d , RRRRRRRRRRR ∑ q ∈Q z ( β, q ) ( cos 2 π (∇ G x , n q ) − ) n q ( y ) ⟨ sin 2 π ( φ, q ) X x d ∗ Y( y, ⋅)⟩ µ β RRRRRRRRRRR ≤ C ∣ y ∣ d − − ε . The proof of the previous estimate relies on the three ingredients: the estimate (2.10), the L ( µ β ) -estimate ∥ X x ∥ L ( µ β ) ≤ C and the estimate (2.13) proved in Step 1. We obtain RRRRRRRRRRR ∑ q ∈Q z ( β, q ) ( cos 2 π (∇ G x , n q ) − ) n q ( y ) ⟨ sin 2 π ( φ, q ) X x d ∗ Y( y, ⋅)⟩ µ β RRRRRRRRRRR≤ ∑ q ∈Q ∣ z ( β, q ) ( cos 2 π (∇ G x , n q ) − ) n q ( y )∣ ∥ sin 2 π ( φ, q ) X x ∥ L ( µ β ) ∥ d ∗ Y( y, ⋅)∥ L ( µ β ) ≤ C ∣ y − x ∣ d − ⋅ ∣ y ∣ d − − ε . Summing the inequality over all the points y ∈ Z d then shows RRRRRRRRRRRRR ∑ y ∈ Z d ⟨ ∑ q ∈Q z ( β, q ) sin 2 π ( φ, q ) ( cos 2 π (∇ G x , n q ) − ) n q ( y ) X x d ∗ Y( y, ⋅)⟩ µ β RRRRRRRRRRRRR ≤ C ∑ y ∈ Z d ∣ y − x ∣ d − ⋅ ∣ y ∣ d − − ε ≤ C ∣ x ∣ d − − ε . The proof of Step 2 is complete.
Step 3. The conclusion.
We use the main result (2.14) of Step 2 and the symmetry of the Helffer-Sj¨ostrandoperator to complete the proof of Lemma 2.1. By the expansion (2.14), we see that to prove (2.3), it issufficient to prove the estimate(2.17) ∑ y ∈ Z d ⟨ Q x ( y, ⋅) X x V( y, ⋅)⟩ µ β = ∑ y ∈ Z d ⟨ Q x ( y, ⋅) X x Y( y, ⋅)⟩ µ β + O ( C ∣ x ∣ d − − ε ) . Since the Helffer-Sj¨ostrand operator L is symmetric, we can write(2.18) ∑ y ∈ Z d ⟨ Q x ( y, ⋅) X x Y( y, ⋅)⟩ µ β = ∑ y ∈ Z d ⟨X x ( y, ⋅) ∂ y Y ⟩ µ β , where the mapping X x ∶ Z d × Ω → R ( d ) is the solution of the Helffer-Sj¨ostrand equation, LX x = Q x X x in Z d × Ω . (2.19) . DECOUPLING THE EXPONENTIALS 153 The objective of this step is to prove the following expansion(2.20) ∑ y ∈ Z d ⟨X x ( y, ⋅) ∂ y Y ⟩ µ β = ∑ y ∈ Z d ⟨X x ( y, ⋅) Q ( y, ⋅) Y ⟩ µ β + O ( C ∣ x ∣ d − − ε ) . The proof is similar to the one written is Steps 1 and 2. With the same arguments as the ones developed inStep 1, one obtains the following upper bound for the function d ∗ X : for each y ∈ Z d ,(2.21) ∥ d ∗ X x ( y, ⋅)∥ L ( µ β ) ≤ C ∣ y − x ∣ d − − ε . Using the same arguments as the ones developed in Step 2, we obtain the inequality
RRRRRRRRRRRR ∑ y ∈ Z d ∑ q ∈Q z ( β, q ) ( cos (∇ G, n q ) − ) n q ( y, φ ) ⟨ d ∗ X x ( y, φ ) sin 2 π ( φ, q ) Y ( φ )⟩ µ β RRRRRRRRRRRR ≤ C ∑ y ∈ Z d ∣ y − x ∣ d − − ε ⋅ ∣ y ∣ d − (2.22) ≤ C ∣ x ∣ d − − ε . Combining the inequalities (2.21) and (2.22) with the formula (2.7) implies the expansion (2.20). We then usethe symmetry of the Helffer-Sj¨ostrand operator a second time to obtain the identity(2.23) ∑ y ∈ Z d ⟨X x ( y, ⋅) Q ( y, ⋅) Y ⟩ µ β = ∑ y ∈ Z d ⟨ Q x ( y, ⋅) X x V ( y, ⋅)⟩ µ β , where the function V is defined as the solution of the Helffer-Sj¨ostrand equation (2.4). Combining theidentities (2.23), (2.20) and (2.18), we obtain the expansion (2.17). This completes the proof of Step 3 and ofLemma 2.1. (cid:3)
3. Decoupling the exponentials
The objective of this section is to remove the exponential terms X x and Y from the computation. Weprove the decorrelation estimate stated in Lemma 3.1 below. The argument makes use of the bounds on theGreen’s function G obtained in Proposition 4.7 of Chapter 3 and on the Green’s function G der , f associated tothe differentiated Helffer-Sj¨ostrand operator proved in Proposition 4.2 of Chapter 5. Before stating the lemma,we record two estimates which are used in its proof: ● We recall the definition of the random variable X x ∶ Z d × Ω → R ( d ) defined as the solution of theHelffer-Sj¨ostrand equation, for each ( z, φ ) ∈ Z d × Ω, LX x ( z, φ ) = ∂ z X x in (2.19) of Section 2; by theinequality (2.21), it satisfies the L ( µ β ) -estimate(3.1) ∥ d ∗ X x ( z, ⋅)∥ L ( µ β ) ≤ C ∣ z − x ∣ d − − ε . For later purposes, we note that the same arguments lead to the estimate(3.2) ∥X x ( z, ⋅)∥ L ( µ β ) ≤ C ∣ z − x ∣ d − − ε ; ● The function V defined in the statement of Lemma 2.1; by the estimate (4.8) of Chapter 4, it satisfiesthe estimate(3.3) ∥ d ∗ V( z, ⋅)∥ L ( µ β ) ≤ C ∣ x ∣ d − − ε . Lemma . One has the following estimate (3.4) cov [ X x , Y ] = ⟨ Y ⟩ µ β ⟨ X ⟩ µ β ∑ y ∈ Z d ⟨ Q x ( y )U( y, ⋅)⟩ µ β + O ( C ∣ x ∣ d − + ε ) , where the function U ∶ Z d × Ω → R ( d ) is the solution of the Helffer-Sj¨ostrand equation (3.5) LU = Q in Z d × Ω .
54 8. PROOF OF THE ESTIMATES IN CHAPTER 4
Proof.
We recall the notations and results introduced in Remarks 4.1, 4.2 and 4.3 of Chapter 4 whichwill be used in the proof. We start from the result of Lemma 2.1 which readscov [ X x , Y ] = ∑ y ∈ Z d ⟨ X x Q x ( y, ⋅)V( y, ⋅)⟩ µ β + O ( C ∣ x ∣ d − − ε ) , where V is the solution of the Helffer-Sj¨ostrand equation, for each ( y, φ ) ∈ Z d × Ω,(3.6)
LV( y, φ ) = Q ( y, φ ) Y ( φ ) . We split the argument into two steps: ● In Step 1, we prove the decorrelation estimate(3.7) ∑ y ∈ Z d ⟨ X x Q x ( y, ⋅)V( y, ⋅)⟩ µ β = ⟨ X x ⟩ µ β ∑ y ∈ Z d ⟨ Q x ( y, ⋅)V( y, ⋅)⟩ µ β + O ( C ∣ x ∣ d − − ε ) . Note that since the measure µ β is invariant under translations, the value ⟨ X x ⟩ µ β does not depend onthe point x . ● In Step 2, we prove the expansion(3.8) ∑ y ∈ Z d ⟨ Q x ( y, ⋅)V( y, ⋅)⟩ µ β = ⟨ Y ⟩ µ β ∑ y ∈ Z d ⟨ Q x ( y, ⋅)U( y, ⋅)⟩ µ β + O ( C ∣ x ∣ d − − ε ) . The expansion (3.4) is a consequence of (3.7) and (3.8)
Step 1.
The expansion (3.7) can be rewritten in terms of the covariance between the random variables X x and Q x ( y )V( y, ⋅) ; it is equivalent to the expansion(3.9) ∑ y ∈ Z d cov [ X x , Q x ( y, ⋅)V( y, ⋅)] = O ( C ∣ x ∣ d − − ε ) . To prove the expansion (3.9), we apply the Helffer-Sj¨ostrand representation formula which reads, for eachpoint y ∈ Z d ,(3.10) cov [ X x , Q x ( y, ⋅)V( y, ⋅)] = ∑ z ∈ Z d ⟨X x ( z, ⋅) ∂ z ( Q x ( y, ⋅)V( y, ⋅))⟩ µ β . Summing over the points y ∈ Z d and performing an integration by parts in the variable y , we deduce that ∑ y ∈ Z d cov [ X x , Q x ( y, ⋅)V( y, ⋅)] = ∑ y,z ∈ Z d ⟨X x ( z, ⋅) ∂ z ( Q x ( y, ⋅)V( y, ⋅))⟩ µ β = ∑ y,z ∈ Z d ⟨X x ( z, ⋅) ∂ z ( n Q x ( y, ⋅) d ∗ V( y, ⋅))⟩ µ β . We split the proof into two substeps: ● In Substep 1.1, we compute the value of ∂ z ( n Q x ( y, ⋅) d ∗ V( y, ⋅)) . We prove the identity (3.46) andthe inequalities (3.47); ● In Substep 1.2, we deduce the estimate (3.7) from Substep 1.1.
Substep 1.1.
We first expand the derivative(3.11) ∂ z ( n Q x ( y, ⋅) d ∗ V( y, ⋅)) = ( ∂ z n Q x ( y, ⋅)) d ∗ V( y, ⋅)·„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„¶ (3.11) −( i ) + n Q x ( y, ⋅) ∂ z d ∗ V( y, ⋅)·„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„¶ (3.11) −( ii ) . The term (3.11)-(i) can be computed explicitly from the definition of the charge n Q x and the identity q = d n q .We obtain ( ∂ z n Q x ( y, φ )) d ∗ V( y, φ ) = ⎛⎝ ∑ q ∈Q π z ( β, q ) ( sin 2 π ( φ, q ) sin 2 π (∇ G x , n q )) n q ( y ) ⊗ q ( z )⎞⎠ d ∗ V( y, φ ) (3.12) = d z ⎛⎝⎛⎝ ∑ q ∈Q π z ( β, q ) ( sin 2 π ( φ, q ) sin 2 π (∇ G x , n q )) n q ( y ) ⊗ n q ( z )⎞⎠ d ∗ V( y, φ )⎞⎠ . . DECOUPLING THE EXPONENTIALS 155 We then estimate the term in the right side of (3.12). To this end, we note that the sum over the charges q ∈ Q can be restricted to the set of charges Q y,z . Using the inequality ∑ q ∈Q y,z e − c √ β ∥ q ∥ ≤ e − c √ β ∣ y − z ∣ establishedin (2.7) of Chapter 2.We use the inequality on the sine of the gradient of the Green’s function: for each charge q ∈ Q such thatthe point y belongs to the support of n q , ∣ sin 2 π (∇ G x , n q )∣ ≤ ∣ π (∇ G x , n q )∣ ≤ C q ∣ x − y ∣ d − . We deduce that
RRRRRRRRRRR ∑ q ∈Q z ( β, q ) sin 2 π ( φ, q ) sin 2 π (∇ G x , n q ) n q ( y ) ⊗ n q ( z )RRRRRRRRRRR ≤ ∑ q ∈Q x,y e − c √ β ∥ q ∥ C q ∣ y − x ∣ d − (3.13) ≤ ∑ q ∈Q x,y C ∥ q ∥ k e − c √ β ∥ q ∥ ∣ y − x ∣ d − ≤ ∑ q ∈Q x,y Ce − c √ β ∥ q ∥ ∣ y − x ∣ d − ≤ Ce − c √ β ∣ y − z ∣ ∣ y − x ∣ d − , where we have reduced the value of the constant c in the third inequality to absorb the algebraic growth of theterm ∥ q ∥ k into the exponential term e − c √ β ∥ q ∥ . Combining the estimate (3.13) with the inequality (3.3) on thecodifferential of the function V , we obtain, for each pair of points z, y ∈ Z d , XXXXXXXXXXX⎛⎝ ∑ q ∈Q z ( β, q ) ( sin 2 π (⋅ , q ) sin 2 π (∇ G x , n q )) n q ( y ) ⊗ n q ( z )⎞⎠ d ∗ V( y, ⋅)XXXXXXXXXXX L ( µ β ) ≤ Ce − c √ β ∣ y − z ∣ ∣ y − x ∣ d − × ∣ y ∣ d − − ε . We now treat the term (3.11)-(ii). To estimate the L ( µ β ) -norm of the map ∂ z d ∗ V( y, φ ) , we start from thedefinition of the map V as the solution of the Helffer-Sj¨ostrand equation (3.6) and apply the derivative ∂ z toboth sides of the identity (3.6). Following the arguments developed at the beginning of Section 4 of Chapter 5,we obtain that the map V der ∶ ( y, z, φ ) → ∂ z V( y, φ ) is the solution of the differentiated Helffer-Sj¨ostrandequation L der V der ( y, z, φ ) = ⎛⎝ ∑ q ∈Q π z ( β, q ) ( sin 2 π ( φ, q ) sin 2 π (∇ G, n q )) q ( y ) ⊗ q ( z )⎞⎠ Y (3.14) + ∑ q ∈Q πz ( β, q ) sin 2 π ( φ, q ) ( d ∗ V , n q ) q ( y ) ⊗ q ( z )− Q ( y, φ ) ⊗ ⎛⎝ Q ( z, φ ) +
12 2 π ∑ q ∈Q z ( β, q ) sin 2 π ( φ, q ) ( cos (∇ G, n q ) − ) q ( z )⎞⎠ Y . We decompose the function V der into three functions, V der , , V der , and V der , according to the three terms inthe right side of (3.14), i.e.,(3.15) ⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩ L der V der , ( y, z, φ ) = ⎛⎝ ∑ q ∈Q z ( β, q ) sin 2 π ( φ, q ) sin 2 π (∇ G, n q ) q ( y ) ⊗ q ( z )⎞⎠ Y , L der V der , ( y, z, φ ) = ∑ q ∈Q z ( β, q ) sin 2 π ( φ, q ) ( d ∗ V , n q ) q ( y ) ⊗ q ( z ) , L der V der , ( y, z, φ ) = − Q ( y, φ ) ⊗ ⎛⎝ Q ( z, φ ) + ∑ q ∈Q z ( β, q ) sin 2 π ( φ, q ) ( cos (∇ G, n q ) − ) q ( z )⎞⎠ Y . We then estimate the three terms V der , , V der , and V der , separately. Estimate for the term V der , . We first express the function V der , in terms of the Green’s matrix G der associated to the differentiated Helffer-Sj¨ostrand operator L der . We obtain, for each pair of points y, z ∈ Z d
56 8. PROOF OF THE ESTIMATES IN CHAPTER 4 and each field φ ∈ Ω, V der , ( y, z, φ ) = ∑ q ∈Q z ( β, q ) ∑ y ,z ∈ Z d G der , sin 2 π (⋅ ,q ) Y ( y, z, φ ; y , z ) sin 2 π (∇ G, n q ) q ( y ) ⊗ q ( z ) (3.16) = ∑ q ∈Q z ( β, q ) ∑ y ,z ∈ Z d d ∗ y d ∗ z G der , sin 2 π (⋅ ,q ) Y ( y, z, φ ; y , z ) sin 2 π (∇ G, n q ) n q ( y ) ⊗ n q ( z ) . Taking the codifferential d ∗ with respect to the variable y on both sides of the identity (3.16), we obtain theformulad ∗ y V der , ( y, z, φ ) = ∑ q ∈Q z ( β, q ) ∑ y ,z ∈ Z d d ∗ y d ∗ y d ∗ y G der , sin 2 π (⋅ ,q ) Y ( y, z, φ ; y , z ) sin 2 π (∇ G, n q ) n q ( y ) ⊗ n q ( z ) . We then use the bound on the triple derivative of the Green’s matrix G der stated in Proposition 4.2 of Chapter 5,the fact that the random variable Y belongs to the space L ( µ β ) and the pointwise estimate (3.13). Weobtain ∥ d ∗ y V der , ( y, z, ⋅)∥ L ( µ β ) ≤ C ∑ y ,z ∈ Z d ∣ y − y ∣ d + − ε + ∣ z − z ∣ d + − ε e − c √ β ∣ y − z ∣ ∣ y ∣ d − (3.17) ≤ C ∑ y ∈ Z d ∣ y − y ∣ d + − ε + ∣ z − y ∣ d + − ε ∣ y ∣ d − . The term in the right side of the inequality (3.17) can be estimated as it was done in (4.17) of Chapter 7; wenote that we have the inequality, for each triplet ( x, y, z ) ∈ Z d × Z d × Z d , (∣ y − y ∣ d + − ε + ∣ z − y ∣ d + − ε ) ≥ c dist Z d (( y, z ) , ( y , y )) d + − ε ≥ c (∣ y − z ∣ + ∣ y + z − y ∣ ) d + − ε ≥ c (∣ y − z ∣ d + − ε + ∣ y + z − z ∣ d + − ε ) , where the notation dist Z d (( y, z ) , ( y , y )) is used to denote the euclidean distance in the lattice Z d betweenthe points ( y, z ) and ( y , y ) , the second inequality is then obtained by computing the orthogonal projection ofthe point ( y, z ) ∈ Z d on the diagonal {( y , y ) ∈ Z d ∶ y ∈ Z d } and the third inequality is obtained by reducingthe value of the constant c . The right side of the inequality (3.17) can be estimated by Proposition 0.3 ofAppendix C and we obtain ∑ y ∈ Z d ∣ y − y ∣ d + − ε + ∣ z − y ∣ d + − ε ∣ y ∣ d − ≤ ∑ y ∈ Z d ∣ y ∣ d − × ∣ y − z ∣ d + − ε + ∣ y + z − y ∣ d + − ε (3.18) ≤ C ∣ y − z ∣ d + max (∣ y ∣ , ∣ z ∣) d − − ε . Plugging the estimate (3.18) into the inequality (3.17), we deduce that ∥ d ∗ y V der , ( y, z, φ )∥ L ( µ β ) ≤ C ∣ y − z ∣ d + max (∣ y ∣ , ∣ z ∣) d − − ε . Multiplying the term by the value n Q x ( y ) and applying the pointwise bound (2.9) shows ∥ n Q x ( y, ⋅) d ∗ y V der , ( y, z, ⋅)∥ L ( µ β ) ≤ C ∣ x − y ∣ d − × ∣ z − y ∣ d + × max (∣ y ∣ , ∣ z ∣) d − − ε . This inequality completes the estimate of the term V der , . Estimate for the term V der , . We first express the function V der , in terms of the Green function G der . Weobtain, for each ( x, y, φ ) ∈ Z d × Z d × Ω, V der , ( y, z, φ ) = − ∑ y ,z ∈ Z d ∑ q ∈Q πz ( β, q ) d ∗ y d ∗ z G der , sin 2 π (⋅ ,q )( d ∗ V ,n q ) ( y, z, φ ; y , z ) n q ( y ) ⊗ n q ( z ) . Taking the codifferential in the variable y on both sides of the identity (3.16), we obtain the formulad ∗ y V der , ( y, z, φ ) = − ∑ y ,z ∈ Z d ∑ q ∈Q πz ( β, q ) d ∗ y d ∗ y d ∗ z G der , sin 2 π (⋅ ,q )( d ∗ V ,n q ) ( y, z, φ ; y , z ) n q ( y ) ⊗ n q ( z ) . . DECOUPLING THE EXPONENTIALS 157 Using Proposition 4.2 of Chapter 5 (for the triple derivative of the Green’s function G der ), we obtain theinequality(3.19) ∥ d ∗ y d ∗ y d ∗ z G der , sin 2 π (⋅ ,q )( d ∗ V ,n q ) ( y, z, ⋅ ; y , z )∥ L ( µ β ) ≤ C ∥( d ∗ V , n q )∥ L ( µ β ) ∣ y − y ∣ d + − ε + ∣ z − z ∣ d + − ε . Using the estimate (3.3) on the L ( µ β ) -norm on the map d ∗ V , we obtain that, for each charge q ∈ Q y , ∥( d ∗ V , n q )∥ L ( µ β ) ≤ C q ∣ y ∣ d − − ε . Summing the estimate (3.19) over all the charges q ∈ Q and using a computation similar to the one performedin (3.13), one obtains ∥ d ∗ V der , ( y, z, ⋅)∥ L ( µ β ) ≤ ∑ y ,z ∈ Z d ∑ q ∈Q y ,z Ce − c √ β ∥ q ∥ ∥( d ∗ V , n q )∥ L ( µ β ) ∥ n q ∥ L ∞ ∣ y − y ∣ d + − ε + ∣ z − z ∣ d + − ε (3.20) ≤ ∑ y ,z ∈ Z d ∑ q ∈Q y ,z C q e − c √ β ∥ q ∥ (∣ y − y ∣ d + − ε + ∣ z − z ∣ d + − ε ) × ∣ y ∣ d − − ε ≤ ∑ y ,z ∈ Z d Ce − c √ β ∣ y − z ∣ (∣ y − y ∣ d + − ε + ∣ z − z ∣ d + − ε ) × ∣ y ∣ d − − ε . Using the exponential decay of the term e − c √ β ∣ y − z ∣ , we can estimate the sum over the variable z of the rightside of the inequality (3.20). We deduce that(3.21) ∥ d ∗ V der , ( y, z, ⋅)∥ L ( µ β ) ≤ ∑ y ∈ Z d C (∣ y − y ∣ d + − ε + ∣ z − y ∣ d + − ε ) × ∣ y ∣ d − − ε . The right side of (3.21) is almost identical to the right side of (3.17) (the only difference is that there is anadditional factor ε in the term ∣ y ∣ d − − ε ) and can be estimated with the same argument. We obtain ∥ n Q x ( y, ⋅) d ∗ y V der , ( y, z, ⋅)∥ L ( µ β ) ≤ C ∣ x − y ∣ d − − ε × ∣ z − y ∣ d + − ε × max (∣ y ∣ , ∣ z ∣) d − − ε . Estimate for the term V der , . This estimate is the most involved of the three terms. We prove that thereexists a map W der , ∶ Z d × Z d × Ω → R ( d )× d which satisfies the identity, for each ( y, z, φ ) ∈ Z d × Z d × Ω,(3.22) V der , ( y, z, φ ) = d z W der , ( y, z, φ ) , as well as the upper bounds(3.23) ∥W der , ( y, z, ⋅)∥ L ( µ β ) ≤ C ∣ y ∣ d − − ε × ∣ z ∣ d − − ε and ∥ d ∗ y W der , ( y, z, ⋅)∥ L ( µ β ) ≤ C ∣ y ∣ d − − ε × ∣ z ∣ d − − ε . To prove the identity (3.22) and the estimates (3.23), we appeal to the parabolic equation, following thestrategy presented in Section 4 of Chapter 5. We recall the notations introduced in this section, and inparticular the Feynman-Kac formula stated in (4.7) and (4.8) of Chapter 5. Applying this formula to theequation defining the map V der , stated in (3.15). We obtain the identity(3.24) V der , ( y, z, φ ) = ∑ y ,z ∈ Z d ∫ ∞ E φ [− Y ( φ t ) P φ ⋅ der ( t, y, z ; y , z ) Q ( y , φ t ) ⊗ Q ( z , φ t )]−
12 2 π ∑ y ,z ∈ Z d ∑ q ∈Q z ( β, q ) ( cos (∇ G, n q ) − ) ∫ ∞ E φ [ sin 2 π ( φ t , q ) Y ( φ t ) P φ ⋅ der ( t, y, z ; y , z ) Q ( y , φ t ) ⊗ q ( z )] , where, given a trajectory ( φ t ) t ≥ of the Langevin dynamics, the map P φ ⋅ der (⋅ , ⋅ , ⋅ ; y , z ) ∶ ( , ∞) × Z d × Z d → R ( d ) denotes the solution of the parabolic system of equations,(3.25) ⎧⎪⎪⎪⎨⎪⎪⎪⎩ ∂ t P φ ⋅ der (⋅ , ⋅ , ⋅ ; y , z ) + (L φ t spat ,x + L φ t spat ,y ) P φ ⋅ der (⋅ , ⋅ , ⋅ ; y , z ) = ( , ∞) × Z d × Z d ,P φ ⋅ der ( , ⋅ , ⋅ ; y , z ) = δ ( y ,z ) in Z d × Z d .
58 8. PROOF OF THE ESTIMATES IN CHAPTER 4
To ease the notation in the rest of the argument, we introduce the following definition. Given a pair of charges q , q ∈ Q and a trajectory of the Langevin dynamics ( φ t ) t ≥ , we let P φ ⋅ q ,q (⋅ , ⋅ , ⋅) ∶ ( , ∞) × Z d × Z d → R ( d ) bethe solution of the parabolic system(3.26) ⎧⎪⎪⎨⎪⎪⎩ ∂ t P φ ⋅ q ,q + (L φ t spat ,x + L φ t spat ,y ) P φ ⋅ q ,q = ( , ∞) × Z d × Z d ,P φ ⋅ q ,q ( , y, z ) = q ( y ) ⊗ q ( z ) . We note that since the operator L φ t spat ,x and L φ t spat ,y commutes, the solution of the equation (3.26) factorizes; onecan write P φ ⋅ q ,q ( t, y, z ) = P φ ⋅ q ( t, y ) ⊗ P φ ⋅ q ( t, z ) , where the maps P φ ⋅ q and P φ ⋅ q are the solutions of the parabolicsystems(3.27) ⎧⎪⎪⎨⎪⎪⎩ ∂ t P φ ⋅ q + L φ t spat P φ ⋅ q = ( , ∞) × Z d ,P φ ⋅ q ( , ⋅) = q in Z d , and ⎧⎪⎪⎨⎪⎪⎩ ∂ t P φ ⋅ q + L φ t spat P φ ⋅ q = ( , ∞) × Z d ,P φ ⋅ q ( , y, z ) = q in Z d . We then use this notation and the definition of the random charge Q stated in (2.1) to rewrite the identity (3.24).We obtain V der , ( y, z, φ ) = − ∑ q ,q ∈Q z ( β, q ) z ( β, q ) sin 2 π (∇ G, n q ) sin 2 π (∇ G, n q ) (3.28) × ∫ ∞ E φ [ cos 2 π ( φ t , q ) cos 2 π ( φ t , q ) Y ( φ t ) P φ ⋅ q ,q ( t, y, z )] dt +
12 2 π ∑ q ,q ∈Q z ( β, q ) z ( β, q ) sin 2 π (∇ G, n q ) ( cos 2 π (∇ G, n q ) − )× ∫ ∞ E φ [ cos 2 π ( φ t , q ) cos 2 π ( φ t , q ) Y ( φ t ) P φ ⋅ q ,q ( t, y, z )] dt. We fix a trajectory ( φ t ) t ≥ of the Langevin dynamics, two points y , z ∈ Z d , a pair of charges ( q , q ) ∈ Q y ×Q z and study the map P φ ⋅ q ,q . More precisely, in view of the decomposition P φ ⋅ q ,q = P φ ⋅ q ⊗ P φ ⋅ q , we study the map P φ ⋅ q for a general charge q ∈ Q and prove the following results:(i) There exist constants C ∶= C ( d ) < ∞ and C q ≤ C e C ∥ q ∥ , for some C ∶= C ( d ) < ∞ , such that eachpoint y ∈ Z d and for each time t ∈ ( , ∞) ,(3.29) ∣ P φ ⋅ q ( t, y )∣ ≤ C q ( βt ) − ε Φ C ( tβ , y − y ) and ∣∇ P φ ⋅ q ( t, y )∣ ≤ C q ( βt ) − ε Φ C ( tβ , y − y ) . We note that here, contrary to the other results presented in the article, the constant C q ,q is allowedto grow exponentially fast in the parameter ∥ q ∥ ; this is caused by the exponential decay of the heatkernel. This growth does not cause problems in the analysis: since the constant C depends onlyon the dimension d , we can use the estimate z ( β, q ) ≤ e − c √ β ∥ q ∥ and set the inverse temperature β large enough so that the exponent c √ β is strictly larger than the constant C in order to absorb thisterm. In particular, from now on and until the estimate (3.45) below, we assume that the constantsare only allowed to depend on the dimension d and keep track of their dependence in the inversetemperature β ;(ii) We prove that there exists a function Q φ ⋅ q ∶ ( , ∞) × Z d → R d such that for each time t ≥ y ∈ Z d one has the identity P φ ⋅ q ( t, y ) = d Q φ ⋅ q ( t, y ) . Additionally, we prove that the function Q φ ⋅ q satisfies the estimates, for each point y ∈ Z d and for each time t ≥ β ,(3.30) ∣ Q φ ⋅ q ( t, y )∣ ≤ C q ( tβ ) ε Φ C ( tβ , y − y ) . To prove the estimate (3.29), we use the results established in Chapter 5. We first express the function P φ ⋅ q interms of the heat kernel P φ ⋅ (defined in (3.25)). We obtain P φ ⋅ q ( t, y ) = ∑ y ′ ∈ Z d d ∗ y ′ P φ ⋅ der ( t, y ; y ′ ) n q ( y ′ ) , and consequently ∇ P φ ⋅ q ( t, y ) = ∑ y ′ ∈ Z d ∇ y d ∗ y ′ P φ ⋅ der ( t, y ; y ′ ) n q ( y ′ ) . . DECOUPLING THE EXPONENTIALS 159 We use the Nash-Aronson estimate and the regularity theory for the heat kernel P φ ⋅ der stated in Propositions 3.5and 3.7 of Chapter 5. We obtain the upper bound(3.31) ∣ P φ ⋅ q ( t, y )∣ ≤ ∥ n q ∥ ∞ ∑ y ′ ∈ supp n q ( βt ) − ε Φ C ( tβ , y − y ′ ) . We use that, by assumption, the point y belongs to the support of the charge n q and the inequality, for eachpoint y ∈ Z d , each point y ′ ∈ supp n q and each time t ∈ ( , ∞) ,Φ C ( t, y − y ′ ) ≤ Φ C ( t, y − y ) exp ( ∣ y − y ∣ C ) ≤ Φ C ( t, y − y ) exp ( diam n q C ) ≤ Φ C ( t, y − y ) exp ( diam qC ) (3.32) ≤ Φ C ( t, y − y ) exp ( ∥ q ∥ C ) , where the first inequality is obtained by an explicit computation using the definition of the function Φ C .Combining the identity (3.31) and the estimate (3.32), we obtain the inequality ∣ P φ ⋅ q ( tβ , y, z )∣ ≤ ∥ n q ∥ ∞ ∑ y ′ ∈ supp n q ( βt ) − ε Φ C ( tβ , y − y ) exp ( ∥ q ∥ C ) (3.33) ≤ ∥ n q ∥ ∞ ∣ supp n q ∣ ( βt ) − ε Φ C ( tβ , y − y ) exp ( ∥ q ∥ C )≤ C q ( βt ) − ε Φ C ( tβ , y − y ) . This completes the proof of the first estimate of (3.29). The second estimate follows similar lines, theonly difference is that we use the regularity estimate stated in Proposition 3.7 of Chapter 5 instead of theNash-Aronson type estimate stated in Proposition 3.5 of Chapter 5.We now focus on the existence of the function Q φ ⋅ q and the estimate (3.30). We first recall the explicitdefinition of the elliptic operator L spat L spat = − β ∆ + β ∑ n ≥ β n (− ∆ ) n + + ∑ q ∈Q ∇ ∗ q ⋅ a q ∇ q . We define the function Q φ ⋅ q to be the solution of the parabolic system(3.34) ⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩ ∂ t Q φ ⋅ q − ( β ∆ − β ∑ n ≥ β n (− ∆ ) n + ) Q φ ⋅ q = − ∑ q ∈Q z ( β, q ) cos 2 π ( φ t , q ) ∇ q P φ ⋅ q n q in ( , ∞) × Z d ,Q φ ⋅ q ( , ⋅) = n q in Z d . Applying the exterior derivative d to both sides of the equation (3.34) and using that this operator commuteswith the Laplacian ∆, we obtain that ⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩ ∂ t d Q φ ⋅ q − ( β ∆ − β ∑ n ≥ β n (− ∆ ) n + ) d Q φ ⋅ q = − ∑ q ∈Q ∇ ∗ q a q ∇ q P φ ⋅ q in ( , ∞) × Z d × Z d , d Q φ ⋅ q ( , ⋅) = q in Z d . Using the definition of the map P φ ⋅ q given in (3.26), we see that the two maps P φ ⋅ q and d Q φ ⋅ q solve the sameparabolic equation with the same initial condition; this implies that they are equal.It remains to prove the estimate (3.30). We denote by K ∶ ( , ∞) × Z d → R ( d )×( d ) the fundamental solutionof the parabolic system ⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩ ∂ t K − ( β ∆ − β ∑ n ≥ β n (− ∆ ) n + ) K = ( , ∞) × Z d ,K ( , ⋅) = δ in Z d .
60 8. PROOF OF THE ESTIMATES IN CHAPTER 4
For the parabolic operator ∂ t − β ∆ + β ∑ n ≥ β n (− ∆ ) n + , a Nash-Aronson estimate and a complete regularitytheory is available: for each integer k ∈ N , there exists a constant C ∶= C ( k, d ) < ∞ such that ∣∇ k K ( t, y )∣ ≤ C ( βt ) k Φ C ( tβ , y ) . We use the Duhamel principle to express the function Q φ ⋅ q in terms of the kernel K . We obtain the formula,for each point y ∈ Z d and each time t > Q φ ⋅ q ( t, y ) = ∑ y ′ ∈ Z d d ∗ K ( t, y − y ′ ) n q ( y ′ )− ∑ y ′ ∈ Z d ∑ q ∈Q ∫ t z ( β, q ) cos 2 π ( φ s , q ) ( n q , d ∗ P φ ⋅ q ( s, ⋅)) K ( t, y − y ′ ) n q ( y ′ ) d s. To estimate the right side of (3.35), we record two inequalities. The first one is obtained by the samecomputation as (3.33) and reads ∑ y ′ ∈ Z d ∣ d ∗ K ( t, y − y ′ ) n q ( y ′ )∣ ≤ ∑ y ′ ∈ Z d C ∥ n q ∥ ∞ ( βt ) − ε Φ C ( tβ , y − y ′ ) (3.36) ≤ C q ( βt ) − ε Φ C ( tβ , y − y ) , where the constant C q satisfies the same exponential growth in the parameter ∥ q ∥ as the one in the right sideof (3.33). For the second inequality, we fix a point y ′ ∈ Z d , a time s >
0, use the Nash-Aronson estimate on thekernel K and the estimate (3.29) on the function P φ ⋅ q . We obtain ∑ q ∈Q ∣ z ( β, q ) cos 2 π ( φ t , q ) ( n q , d ∗ P φ ⋅ q ( s, ⋅)) K ( t − s, y − y ′ ) n q ( y ′ )∣ (3.37) ≤ ∑ q ∈Q y ′ C q e − c √ β ∥ q ∥ Φ C ( t − sβ , y − y ′ ) ∥∇ P φ ⋅ q ( s, ⋅)∥ L ( supp n q ) ≤ ∑ q ∈Q y ′ C q e − c √ β ∥ q ∥ Φ C ( t − sβ , y − y ′ ) ∥ Φ C ( sβ , ⋅)∥ L ( supp n q ) ( sβ ) − ε ∧ . With a computation similar to the one performed in (3.32), we obtain the inequality, for each charge q ∈ Q y ′ ,(3.38) ∥ Φ C ( sβ , ⋅)∥ L ( supp n q ) ≤ ∥ Φ C ( sβ , y ′ ) exp ( ∥ q ∥ C )∥ L ( supp n q ) ≤ C q exp ( ∥ q ∥ C ) Φ C ( sβ , y ′ ) . Combining the inequalities (3.37) and (3.38), we have obtained(3.39) ∑ q ∈Q ∣ z ( β, q ) cos 2 π ( φ t , q ) ( n q , d ∗ P φ ⋅ q ( s, ⋅)) K ( t − s, y − y ′ ) n q ( y ′ )∣≤ Φ C ( t − sβ , y − y ′ ) Φ C ( sβ , y ′ )( sβ ) − ε ∧ ∑ q ∈Q y ′ C q e − c √ β ∥ q ∥ . ·„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„¶ (3.39) −( i ) By choosing the inverse temperature β large enough, we obtain that the sum (3.39)-(i) is bounded from aboveby a constant C depending only on d ; we have proved(3.40) RRRRRRRRRRR ∑ q ∈Q z ( β, q ) cos 2 π ( φ t , q ) ( n q , d ∗ P φ ⋅ q ( s, ⋅)) K ( t − s, y − y ′ ) n q ( y ′ )RRRRRRRRRRR ≤ C Φ C ( t − sβ , y − y ′ ) Φ C ( sβ , y ′ ) s − ε ∧ . We then use the inequality, for some constant ̃ C > C ,(3.41) ∑ y ′ ∈ Z d Φ C ( t − sβ , y − y ′ ) Φ C ( sβ , y ′ ) ≤ ̃ C Φ ̃ C ( tβ , y ) . . DECOUPLING THE EXPONENTIALS 161 The inequality (3.41) is a convolution property for the discrete heat kernel and can be verified by an explicitcomputation using the formula for the map Φ C . A combination of the inequalities (3.40) and (3.41) yields(3.42) ∑ y ′ ∈ Z d ∑ q ∈Q ∣ z ( β, q ) cos 2 π ( φ t , q ) ( n q , d ∗ P φ ⋅ q ( s, ⋅)) K ( t − s, y − y ′ ) n q ( y ′ )∣ ≤ C Φ C ( tβ , y )( sβ ) − ε ∧ . Integrating the inequality (3.42) in the time interval [ , t ] and using the assumption t ≥ β , we obtain theestimate ∫ t ∑ y ′ ∈ Z d ∑ q ∈Q ∣ z ( β, q ) cos 2 π ( φ t , q ) ( n q , d ∗ P φ ⋅ q ( s, ⋅)) K ( t − s, y − y ′ ) n q ( y ′ )∣ ≤ ∫ t C Φ C ( tβ , y )( sβ ) − ε ∧ s ≤ C ( tβ ) ε Φ C ( tβ , y ) . This completes the proof of the estimate of (3.30).We are now in position to define the function W der , so that it satisfies the identity (3.22) and theinequalities (3.23). We use the identity (3.28) and define the map W der , by the formula W der , ( y, z, φ ) = − ∑ q ,q ∈Q z ( β, q ) z ( β, q ) sin 2 π (∇ G, n q ) sin 2 π (∇ G, n q ) (3.43) × ∫ ∞ E φ [ cos 2 π ( φ t , q ) cos 2 π ( φ t , q ) Y ( φ t ) P φ ⋅ q ( t, y ) ⊗ Q φ ⋅ q ( t, z )] dt +
12 2 π ∑ q ,q ∈Q z ( β, q ) z ( β, q ) sin 2 π (∇ G, n q ) ( cos 2 π (∇ G, n q ) − )× ∫ ∞ E φ [ cos 2 π ( φ t , q ) cos 2 π ( φ t , q ) Y ( φ t ) P φ ⋅ q ( t, y ) ⊗ Q φ ⋅ q ( t, z )] dt. Applying the exterior derivative d in the z -variable and using the identity P φ ⋅ q = d Q φ ⋅ q , we obtain theidentity (3.28). There only remains to prove the inequalities (3.23). To this end, we first take the L ( µ β ) -normon both sides of the identity (3.43) and use the following facts: the absolute value of the cosine is boundedby 1, the random variable Y belongs to the space L ( µ β ) , the Langevin dynamics is invariant under the Gibbsmeasure µ β and the functions P φ ⋅ q and Q φ ⋅ q are bounded from above by the right sides of (3.29) and (3.30).We obtain ∥W der , ( y, z, ⋅)∥ L ( µ β ) ≤ ∑ y ′ ,z ′ ∈ Z d ∑ q ∈Q y ′ ,q ∈Q z ′ ∣ z ( β, q ) z ( β, q ) sin 2 π (∇ G, n q ) sin 2 π (∇ G, n q )∣ (3.44) × ∫ ∞ ( tβ ) − + ε Φ C ( tβ , y − y ′ ) Φ C ( tβ , z − z ′ ) dt +
12 2 π ∑ y ′ ,z ′ ∈ Z d ∑ q ∈Q y ′ ,q ∈Q z ′ ∣ z ( β, q ) z ( β, q ) sin 2 π (∇ G, n q ) ( cos 2 π (∇ G, n q ) − )∣× ∫ ∞ ( tβ ) − + ε Φ C ( tβ , y − y ′ ) Φ C ( tβ , z − z ′ ) dt. We use the estimates, for each point y ′ ∈ Z d and each charge q ∈ Q y ′ ∣ sin 2 π (∇ G, n q )∣ ≤ C q ∣ y ′ ∣ d − and ∣ cos 2 π (∇ G, n q ) − ∣ ≤ C q ∣ y ′ ∣ d − .
62 8. PROOF OF THE ESTIMATES IN CHAPTER 4
Allowing the constants to depend on the inverse temperature β , we obtain that ∥W der , ( y, z, ⋅)∥ L ( µ β ) ≤ ∑ y ′ ,z ′ ∈ Z d ∑ q ∈Q y ′ ,q ∈Q z ′ e − c √ β (∥ q ∥ +∥ q ∥ ) C q C q ∣ y ′ ∣ d − ∣ z ′ ∣ d − (∣ y − y ′ ∣ d − − ε + ∣ z − z ′ ∣ d − − ε ) (3.45) + ∑ y ′ ,z ′ ∈ Z d ∑ q ∈Q y ′ ,q ∈Q z ′ e − c √ β (∥ q ∥ +∥ q ∥ ) C q C q ∣ y ′ ∣ d − ∣ z ′ ∣ d − (∣ y − y ′ ∣ d − − ε + ∣ z − z ′ ∣ d − − ε )≤ ∑ y ′ ,z ′ ∈ Z d C ∣ y ′ ∣ d − ∣ z ′ ∣ d − (∣ y − y ′ ∣ d − − ε + ∣ z − z ′ ∣ d − − ε )+ ∑ y ′ ,z ′ ∈ Z d C ∣ y ′ ∣ d − ∣ z ′ ∣ d − (∣ y − y ′ ∣ d − − ε + ∣ z − z ′ ∣ d − − ε ) , where in the second inequality we used the exponential decays of the term e − c √ β (∥ q ∥ +∥ q ∥ ) to absorb thealgebraic growth of the constants C q and C q . The right hand side of the estimate (3.45) is estimated bynoting that ∣ z ′ ∣ d − ≥ ∣ z ∣ d − and by using the inequality a + b ≥ √ ab . We obtain ∥W der , ( y, z, ⋅)∥ L ( µ β ) ≤ ∑ y ′ ,z ′ ∈ Z d C ∣ y ′ ∣ d − ∣ z ′ ∣ d − (∣ y − y ′ ∣ d − − ε + ∣ z − z ′ ∣ d − − ε )≤ ∑ y ′ ,z ′ ∈ Z d C ∣ y ′ ∣ d − ∣ z ′ ∣ d − ∣ y − y ′ ∣ d − − ε ∣ z − z ′ ∣ d − − ε ≤ ⎛⎝ ∑ y ′ ∈ Z d C ∣ y ′ ∣ d − ∣ y − y ′ ∣ d − − ε ⎞⎠ ( ∑ z ′ ∈ Z d C ∣ z ′ ∣ d − ∣ z − z ′ ∣ d − − ε )≤ C ∣ y ∣ d − − ε ∣ z ∣ d − − ε . This completes the estimate for the map W der , stated in (3.23). To estimate the second estimate (3.23),involving the codifferential d ∗ y is similar and we only give an outline of the argument. First, by taking thecodifferential on both sides of the identity (3.43), we obtain the formulad ∗ y W der , ( y, z, φ ) = − ∑ q ,q ∈Q z ( β, q ) z ( β, q ) sin 2 π (∇ G, n q ) sin 2 π (∇ G, n q )× ∫ ∞ E φ [ cos 2 π ( φ t , q ) cos 2 π ( φ t , q ) Y ( φ t ) d ∗ y P φ ⋅ q ( t, y ) ⊗ Q φ ⋅ q ( t, z )] dt +
12 2 π ∑ q ,q ∈Q z ( β, q ) z ( β, q ) sin 2 π (∇ G, n q ) ( cos 2 π (∇ G, n q ) − )× ∫ ∞ E φ [ cos 2 π ( φ t , q ) cos 2 π ( φ t , q ) Y ( φ t ) d ∗ y P φ ⋅ q ( t, y ) ⊗ Q φ ⋅ q ( t, z )] dt. One can then rewrite the same argument, and use the second estimate of (3.29) on the map ∇ P φ ⋅ q (whichprovides an upper bound for the map d ∗ P φ ⋅ q since the codifferential is a linear functional of the gradient)instead on the first estimate of (3.29) on the map P φ ⋅ q to obtain the result. The estimate of the term W der , stated in (3.23) is complete. We conclude the estimate of the term V der , by multiplying the term by the value n Q x ( y, φ ) . We obtain n Q x ( y, φ ) d ∗ y V der , ( y, z, φ ) = n Q x ( y, φ ) d z d ∗ y W der , ( y, z, φ ) = d z ( n Q x ( y, φ ) d ∗ y W der , ( y, z, φ )) , where in the second equality, we used that since the exterior derivative d z only acts on the z -variable, itcommutes with the codifferential d ∗ y (which only acts on the y -variable). Applying the pointwise bound (2.9)shows the inequality ∥ n Q x ( y, ⋅) d ∗ y W der , ( y, z, ⋅)∥ L ( µ β ) ≤ ∥ n Q x ( y, ⋅)∥ L ∞ ( µ β ) ∥ d ∗ y W der , ( y, z, ⋅)∥ L ( µ β ) ≤ C ∣ y − x ∣ d − ∣ y ∣ d − − ε ∣ z ∣ d − − ε . . DECOUPLING THE EXPONENTIALS 163 Conclusion of Substep 1.1.
Collecting the results proved in this step, we have obtained the identity, foreach pair of points ( y, z ) ∈ Z d , ∂ z ( n Q x ( y, ⋅) d ∗ V( y, ⋅)) = ( ∂ z n Q x ( y, ⋅)) d ∗ V( y, ⋅) + n Q x ( y, ⋅) ∂ z d ∗ V( y, ⋅) (3.46) = ( ∂ z n Q x ( y, ⋅)) d ∗ V( y, ⋅) + n Q x ( y, ⋅) d ∗ y V der , ( y, z, ⋅)+ n Q x ( y, ⋅) d ∗ y V der , ( y, z, ⋅) + d z ( n Q x ( y, ⋅) d ∗ y W der , ( y, z, ⋅)) , with the estimates(3.47) ⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩ ∥( ∂ z n Q x ( y, ⋅)) d ∗ V( y, ⋅)∥ L ( µ β ) ≤ Ce − c √ β ∣ y − z ∣ ∣ y − x ∣ d − × ∣ y ∣ d − − ε , ∥ n Q x ( y, ⋅) d ∗ y V der , ( y, z, ⋅)∥ L ( µ β ) ≤ C ∣ x − y ∣ d − − ε × ∣ z − y ∣ d + − ε × max (∣ y ∣ , ∣ z ∣) d − − ε , ∥ n Q x ( y, ⋅) d ∗ y V der , ( y, z, ⋅)∥ L ( µ β ) ≤ C ∣ x − y ∣ d − − ε × ∣ z − y ∣ d + − ε × max (∣ y ∣ , ∣ z ∣) d − − ε , ∥ n Q x ( y, ⋅) d ∗ y W der , ( y, z, ⋅)∥ L ( µ β ) ≤ C ∣ y − x ∣ d − ∣ y ∣ d − − ε ∣ z ∣ d − − ε . Substep 1.2.
We prove the covariance estimate (3.9) rewritten below:(3.48) ∑ y ∈ Z d cov [ X x , Q x ( y, ⋅)V( y, ⋅)] = O ( C ∣ x ∣ d − − ε ) . By the Helffer-Sj¨ostrand representation formula we have, for each point y ∈ Z d ,(3.49) cov [ X x , Q x ( y, ⋅)V( y, ⋅)] = ∑ z ∈ Z d ⟨X x ( z, ⋅) ∂ z ( Q x ( y, ⋅)V( y, ⋅))⟩ µ β , where X x is the solution of the Helffer-Sj¨ostrand equation, for each pair ( z, φ ) ∈ Z d × Ω, LX x ( z, φ ) = ∂ z X x ( φ ) . We recall the upper bounds (3.1) and (3.2) on the L ( µ β ) -norm of the map X x and the exterior derivatived ∗ X x : for each point z ∈ Z d ,(3.50) ∥X x ( z, ⋅)∥ L ( µ β ) ≤ C ∣ z − x ∣ d − − ε and ∥ d ∗ X x ( z, ⋅)∥ L ( µ β ) ≤ C ∣ z − x ∣ d − − ε . Using the formula (3.49), we can rewrite the expansion (3.48) ∑ y ∈ Z d cov [ X x , Q x ( y, ⋅)V( y, ⋅)] = ∑ y,z ∈ Z d ⟨X x ( z, ⋅) ∂ z ( Q x ( y, ⋅)V( y, ⋅))⟩ µ β (3.51) = ∑ y,z ∈ Z d ⟨X x ( z, ⋅) ∂ z ( n Q x ( y, ⋅) d ∗ V( y, ⋅))⟩ µ β . We combine the identities (3.46) and (3.51) and write ∑ y ∈ Z d cov [ X x , Q x ( y, ⋅)V( y, ⋅)]= ∑ y,z ∈ Z d ⟨X x ( z, ⋅) ( ∂ z n Q x ( y )) d ∗ V( y, ⋅)⟩ µ β + ∑ y,z ∈ Z d ⟨X x ( z, ⋅) n Q x ( y ) d ∗ y V der , ( y, z, ⋅)⟩ µ β + ∑ y,z ∈ Z d ⟨X x ( z, ⋅) n Q x ( y ) d ∗ y V der , ( y, z, ⋅)⟩ µ β + ∑ y,z ∈ Z d ⟨ d ∗ X x ( z, ⋅) ( n Q x ( y ) d ∗ y W der , ( y, z, ⋅))⟩ µ β .
64 8. PROOF OF THE ESTIMATES IN CHAPTER 4
We use the estimates (3.50) on the function X x and the estimates (3.47). We obtain ∑ y ∈ Z d cov [ X x , Q x ( y, ⋅)V( y, ⋅)] ≤ ∑ y,z ∈ Z d C ∣ z − x ∣ d − − ε e − c √ β ∣ y − z ∣ ∣ y − x ∣ d − × ∣ y ∣ d − − ε (3.52) + ∑ y,z ∈ Z d C ∣ z − x ∣ d − − ε ∣ x − y ∣ d − − ε × ∣ z − y ∣ d + − ε × max (∣ y ∣ , ∣ z ∣) d − − ε + ∑ y,z ∈ Z d C ∣ z − x ∣ d − − ε ∣ y − x ∣ d − ∣ y ∣ d − − ε ∣ z ∣ d − − ε . We estimate the three terms in the right side of (3.52) separately. For the first term, we have(3.53) ∑ y,z ∈ Z d C ∣ z − x ∣ d − − ε Ce − c √ β ∣ y − z ∣ ∣ y − x ∣ d − × ∣ y ∣ d − − ε ≤ ∑ y ∈ Z d C ∣ y − x ∣ d − − ε × ∣ y ∣ d − − ε ≤ C ∣ x ∣ d − − ε . For the second term, we use the inequality max (∣ y ∣ , ∣ z ∣) ≥ ∣ y ∣ and write ∑ y,z ∈ Z d C ∣ z − x ∣ d − − ε ∣ x − y ∣ d − − ε × ∣ z − y ∣ d + − ε × max (∣ y ∣ , ∣ z ∣) d − − ε (3.54) ≤ ∑ y ∈ Z d [ C ∣ x − y ∣ d − − ε ∣ y ∣ d − − ε ∑ z ∈ Z d ∣ z − x ∣ d − − ε ∣ z − y ∣ d + − ε ]≤ ∑ y ∈ Z d C ∣ x − y ∣ d − − ε ∣ y ∣ d − − ε ∣ x − y ∣ d − − ε ≤ C ∣ x ∣ d − − ε . For the third term, we have ∑ y,z ∈ Z d C ∣ z − x ∣ d − − ε C ∣ y − x ∣ d − ∣ y ∣ d − − ε ∣ z ∣ d − − ε = ⎛⎝ ∑ y ∈ Z d C ∣ y − x ∣ d − ∣ y ∣ d − − ε ⎞⎠ ( ∑ z ∈ Z d C ∣ z − x ∣ d − − ε ∣ z ∣ d − − ε ) (3.55) ≤ C ∣ x ∣ d − − ε C ∣ x ∣ d − − ε ≤ C ∣ x ∣ d − − ε , where we used the inequality 2 d − ≥ d −
1, valid in dimension larger than 3. Combining the inequali-ties (3.52), (3.53), (3.54) and (3.55) completes the proof of Step 1.
Step 2.
To complete the proof of Lemma 3.1, there remains to prove the expansion (3.8) which is restatedbelow(3.56) ∑ y ∈ Z d ⟨ Q x ( y, ⋅)V( y, ⋅)⟩ µ β = ⟨ Y ⟩ µ β ∑ y ∈ Z d ⟨ Q x ( y, ⋅)U( y, ⋅)⟩ µ β + O ( C ∣ x ∣ d − − ε ) , where the functions V and U are respectively defined in (3.5) and (3.6). The strategy of the proof relies on thesymmetry of the Helffer-Sj¨ostrand operator L ; if we let U x the solution of the equation LU x = Q x in Z d × Ω,then we have the identities ∑ y ∈ Z d ⟨ Q x ( y, ⋅)V( y, ⋅)⟩ µ β = ∑ y ∈ Z d ⟨U x ( y, ⋅) Q ( y, ⋅) Y ⟩ µ β and ∑ y ∈ Z d ⟨ Q x ( y, ⋅)U( y, ⋅)⟩ µ β = ∑ y ∈ Z d ⟨U x ( y, ⋅) Q ( y, ⋅)⟩ µ β . Using these identities, we see that the expansion (3.56) is equivalent to ∑ y ∈ Z d ⟨U x ( y, ⋅) Q ( y, ⋅) Y ⟩ µ β = ⟨ Y ⟩ µ β ∑ y ∈ Z d ⟨U x ( y, ⋅) Q ( y, ⋅)⟩ µ β + O ( C ∣ x ∣ d − − ε ) . The proof of this result is similar to the proof written in Step 1, and in fact simpler since we do not have totreat the term V der , in (3.15); we omit the details. (cid:3) . USING THE SYMMETRY AND ROTATION INVARIANCE OF THE DUAL VILLAIN MODEL 165
4. Using the symmetry and rotation invariance of the dual Villain model
This section is devoted of some properties of the discrete convolution of the discrete Green’s function onthe lattice Z d . We recall the definition of the group H of the lattice-preserving maps introduced in Section 1of Chapter 2. Lemma . Fix four integers j, j , k, k ∈ { , . . . , d } and let F ∶ Z d → R be the function F j,k,j ,k ( x ) ∶= ∑ y,κ ∈ Z d ∇ j G ( y )∇ k G ( x − y − κ )∇ j ∇ k G ( κ ) . Then there exists a ( − d ) -homogeneous map J j,k,j ,k ∶ R d ∖ { } → R such that for any exponent ε > , thereexists a constant C ∶= C ( d, ε ) < ∞ satisfying F j,k,j ,k ( x ) = J j,k,j ,k ( x ) + O ( C ∣ x ∣ d − − ε ) . The function J i,j,i ,j is given by the formula J j,k,j ,k ( x ) = ∫ ∞ t − d + K j,k,j ,k ( xt ) dt, where K ∶ R d → R is defined as the inverse Fourier transform K j,k,j ,k ( x ) = ( π ) d ∫ R d ξ j ξ k ξ j ξ k e −∣ ξ ∣ e ix ⋅ ξ d ξ. Remark . Fix two integers j, j ∈ { , . . . , d } and let F j,j ∶ Z d → R be the map F j,j ( x ) = ∑ y ∈ Z d ∇ j G ( y )∇ j G ( x − y ) . An adaptation of the proof of Lemma 4.1 shows the identity F j,j ( x ) = ∫ ∞ t − d + K j,j ( xt ) dt + O ( C ∣ x ∣ d − − ε ) , where the map K j,j is defined as the inverse Fourier transform K j,j ( x ) = ( π ) d ∫ R d ξ j ξ j e −∣ ξ ∣ e ix ⋅ ξ d ξ. Proof.
The proof uses an explicit formula for the Green’s function and relies on the discrete Fouriertransform. Let us try to construct the Green’s function G by Fourier transform. Given a function F ∶ Z d → R which decays sufficiently fast at infinity, we define its discrete Fourier transform according to the formula, foreach ξ = ( ξ , . . . , ξ d ) ∈ R d , ˆ F ( ξ ) = ∑ z ∈ Z d F ( z ) e − iξ ⋅ z . Then we have ̂ ∆ F ( ξ ) = ∑ dl = ( cos ξ l − ) ˆ F ( ξ ) . Applying this formula to the Green’s function G , we formallyobtain that ˆ G = ( ∑ dl = ( cos ξ l − )) − . Applying the inverse Fourier transform, we formally obtain the formula,for each z ∈ Z d ,(4.1) G ( z ) = ( π ) d ∫ [− π,π ] d e iξ ⋅ z ∑ dl = ( − cos ξ l ) d ξ. One can then prove the formula (4.1) rigorously. First, it is clear that the integral (4.1) converges absolutelyin dimension larger than 3 and that it is bounded as a function of the variable z ; one can then verify by anexplicit computation that the discrete Laplacian in the variable z of the integral (4.1) is equal to the Dirac δ .This argument provides a rigorous justification of the identity (4.1).Once the formula (4.1) is established, we obtain the identity for the gradient of the function G : for eachinteger j ∈ { , . . . , d } , ∇ j G ( z ) = ( π ) d ∫ [− π,π ] d ( e iξ j − ) e iξ ⋅ z ∑ dl = ( − cos ξ l ) d ξ.
66 8. PROOF OF THE ESTIMATES IN CHAPTER 4
Using that the discrete Fourier transform turns discrete convolution into products, we obtain the formula(4.2) F j,k,j ,k ( x ) = ( π ) d ∫ [− π,π ] d ( e iξ j − ) ( e iξ k − ) ( e iξ j − ) ( e iξ k − ) e iξ ⋅ x ( ∑ dl = ( − cos ξ l )) d ξ. We use the identity, for any ξ ∈ [− π, π ] d ∖ { } ,1 ( ∑ dl = ( cos ξ l − )) = ∫ ∞ e − t ( ∑ dl = ( − cos ξ l )) d t and apply Fubini’s theorem to rewrite the identity (4.2), F j,k,j ,k ( x ) = ( π ) d ∫ ∞ ∫ [− π,π ] d ( e iξ j − ) ( e iξ k − ) ( e iξ j − ) ( e iξ k − ) e iξ ⋅ x e − t ( ∑ dl = ( − cos ξ l )) d ξ d t. To ease the notation, we denote by(4.3) u ( t, x ) ∶= ∫ [− π,π ] d ( e iξ j − ) ( e iξ k − ) ( e iξ j − ) ( e iξ k − ) e iξ ⋅ x e − t ( ∑ dl = ( − cos ξ l )) d ξ. We prove the two following properties on the map u :(1) For any exponent α >
0, each integer m ∈ N , there exists a constant C m,α ∶= C m,α ( m, α, d ) < ∞ suchthat for any λ ≥
1, any time t ≤ x ∈ R d such that ∣ x ∣ = λx ∈ Z d ,(4.4) ∣ u ( λ − α t, λx )∣ ≤ C m,α λ − m ;(2) For any time t ∈ ( , ∞) , any point x ∈ R d and any λ ∈ ( , ∞) such that √ tλ ≥ λ d + u ( λ t, λx ) = ∫ R d ξ j ξ k ξ j ξ k e iξ ⋅ x e − t ∣ ξ ∣ d ξ + O ( Cλt d + ) . We first prove (4.4). Fix a point x ∈ R d such that ∣ x ∣ = P ( t, x ) ∶= ∫ [− π,π ] d e iξ ⋅ x e − t ( ∑ dl = ( − cos ξ l )) d ξ. Note that u ( t, x ) = ∇ j ∇ j ∇ k ∇ k P ( t, x ) . Using that the discrete gradients are bounded operators, one sees that to prove (4.4) it is sufficient to provethe inequality(4.7) ∣ P ( λ − α t, λx )∣ ≤ C k,α λ − k . Consider the identity (4.6) and perform the change of variables ξ ∶= λξ (4.8) λ d P ( λ − α t, λx ) = ∫ [− λπ,λπ ] d e iξ ⋅ x e − tλ − α ( ∑ dl = ( − cos ξlλ )) d ξ. We write x = ( x , . . . , x d ) to denote the components of the vector x and assume without loss of generality that ∣ x ∣ ≥ d (since we have assumed ∣ x ∣ = g ∶ R d → R and g λ,t ∶ R d → R be the mappings defined by theformulas g ( ξ ) ∶= ( d ∑ l = ( − cos ξ l )) and g λ,t ( ξ ) ∶= tλ − α ( d ∑ l = ( − cos ξ l λ )) . Note that the function g is analytic and that the first order term of its Taylor expansion around the point 0 isgiven by g ( ξ ) = ∣ ξ ∣ + O ( C ∣ ξ ∣ ) . From these observations, one deduces the upper bound: for any integer n ∈ N there exists a constant C n ∶= C n ( d, n ) < ∞ such that for any point ξ ∈ [− π, π ] d ,(4.9) ∣ ∂ n g ( ξ )∣ ≤ C n ∣ ξ ∣ max ( , − n ) , where the symbol ∂ n denotes the partial derivative with respect to the variable ξ iterated n -times. Aconsequence of the estimate (4.9) is the following inequality: for any integer n ∈ N ∗ and any ξ ∈ [− λπ, λπ ] d ,(4.10) ∣ ∂ n g λ,t ( ξ )∣ ≤ C n tλ − α λ n ( ∣ ξ ∣ λ ) max ( , − n ) . . USING THE SYMMETRY AND ROTATION INVARIANCE OF THE DUAL VILLAIN MODEL 167 For later use, we rewrite the inequality (4.10) in the following form ∣ ∂ n g λ,t ( ξ )∣ ≤ C n ( tλ − α ) min ( , n ) λ max ( ,n − ) ( t λ − α ∣ ξ ∣) max ( , − n ) (4.11) ≤ C n λ α min ( , n )+ max ( ,n − ) ( t λ − α ∣ ξ ∣) max ( , − n ) ≤ C n λ − αn ( t λ − α ∣ ξ ∣) max ( , − n ) . where in the second line we have used the inequality t ≤ α min ( , n ) + max ( , n − ) ≥ αn (since α ≤ ξ → e ix ⋅ ξ and e − g λ,t are 2 λπ -periodic (by the assumption λx ∈ Z d ),we perform k consecutive integration by parts with respect to the variable ξ in the right side of (4.8). Weobtain(4.12) λ d P ( λ − α t, λx ) = ∫ [− λπ,λπ ] d ( − ix ) k e iξ ⋅ x ∂ k e − g λ,t ( ξ ) d ξ. Applying Fa`a di Bruno’s formula, we write(4.13) ∂ k e − g λ,t ( ξ ) = ∑ k ! m ! 1! m m ! 2! m ⋯ m k ! k ! m k ⋅ e − g λ,t ( ξ ) ⋅ k ∏ j = (− g ( j ) λ,t ( ξ )) m j , where the sum runs over over all n -tuples of nonnegative integers ( m , ..., m k ) satisfying the constraint(4.14) m + m + . . . + km k = k. We combine the identity (4.13) with the estimate (4.11) and obtain ∣ ∂ k e − g ( ξ )∣ ≤ ∑ k ! m ! 1! m m ! 2! m ⋯ m k ! k ! m k ⋅ e − g λ,t ( ξ ) ⋅ k ∏ j = ∣ g ( j ) ( ξ )∣ m j (4.15) ≤ ∑ k ! m ! 1! m m ! 2! m ⋯ m k ! k ! m k ⋅ e − g λ,t ( ξ ) ⋅ k ∏ j = ( C j λ − αj ( t λ − α ∣ ξ ∣) max ( , − j ) ) m j ≤ ∑ k ! ∏ kj = C m j j m ! 1! m m ! 2! m ⋯ m k ! k ! m k ⋅ e − g λ,t ( ξ ) λ − α ∑ kj = jm j ( t λ − α ∣ ξ ∣) ∑ kj = m j max ( , − j ) To treat the right side of the inequality (4.15), we use the identity (4.14) and note that since all the constantswhich appear in this term depend only on the integer k and the dimension d , we may replace them by thenotation C k . We obtain(4.16) ∣ ∂ k e − g λ,t ( ξ )∣ ≤ C k λ − αk ⋅ e − g λ,t ( ξ ) ∑ ( t λ − α ∣ ξ ∣) ∑ kj = m j max ( , − j ) . By the identity (4.14), we have(4.17) k ∑ j = m j max ( , − j ) ≤ k ∑ j = m j ≤ k ∑ j = jm j ≤ k. Combining the inequality (4.17) with the identity r κ ≤ + r k , valid for any r ∈ [ , ∞) and any κ ∈ { , . . . , k } ,we can write(4.18) ( t λ − α ∣ ξ ∣) ∑ kj = m j max ( , − j ) ≤ + ( t λ − α ∣ ξ ∣) k . Using the estimate (4.18) and the fact that the sum in the right side of (4.16) contains at most C k terms, wededuce that(4.19) ∣ ∂ k e − g λ,t ( ξ )∣ ≤ C k λ − αk ⋅ e − g λ,t ( ξ ) ( + ( t λ − α ∣ ξ ∣) k ) . We then use that there exists a universal constant C such that, for any r ∈ [− π, π ] d , 1 − cos r ≤ r C to obtainthe following estimate on the map e − g λ,t : for any ξ ∈ [− λπ, λπ ] d , e − g λ,t ( ξ ) ≤ e − tλ − α ∣ ξ ∣ C .
68 8. PROOF OF THE ESTIMATES IN CHAPTER 4
Consequently, there exists a constant C k which depends only on the integer k such that, for any ξ ∈ [− λπ, λπ ] d ,(4.20) ( + ( t λ − α ∣ ξ ∣) k ) e − g λ,t ( ξ ) ≤ ( + ( t λ − α ∣ ξ ∣) k ) e − tλ − α ∣ ξ ∣ C ≤ C k e − tλ − α ∣ ξ ∣ C . Combining (4.19) with (4.20), we have obtained(4.21) ∣ ∂ k e − g λ,t ( ξ )∣ ≤ C k λ − αk e − tλ − α ∣ ξ ∣ C ≤ C k λ − αk . We now complete the proof of (4.7) using (4.21). We let m be the integer which appears in (4.7) and k be thesmallest integer such that αk ≥ m . We consider the identity (4.12), use the estimate (4.21) and the assumption ∣ x ∣ ≥ d . We obtain ∣ P ( λ − α t, λx )∣ ≤ λ − d ∫ [− λπ,λπ ] d d k C k λ − αk ≤ C k λ − αk ≤ C k λ − m , since the integer k is chosen such that it depends only on m and α , we have obtained (4.7). The proof of (4.4)is complete.It remains to prove (4.5). Consider the identity (4.3) and perform the change of variable ξ ∶= λt ξ ,(4.22) ( √ tλ ) d + u ( λ t, λx )= ∫ [− √ tλπ, √ tλπ ] d ( √ tλ ) ( e i ξj √ tλ − ) ( e i ξk √ tλ − ) ( e i ξj √ tλ − ) ( e i ξk √ tλ − ) e iξ ⋅ x √ t e − tλ ( ∑ dl = ( − cos ξl √ tλ )) d ξ. Note that, by using the assumption t ≥ λ by √ tλ in (4.22), it is sufficient, in order toprove (4.5), to prove the following expansion: for any λ ∈ ( , ∞) and any point x ∈ R d ,(4.23) ∫ [− λπ,λπ ] d λ ( e i ξjλ − ) ( e i ξkλ − ) ( e i ξj λ − ) ( e i ξk λ − ) e iξ ⋅ x e − λ ( ∑ di = ( − cos ξiλ )) d ξ = ∫ R d ξ j ξ k ξ j ξ k e iξ ⋅ x e −∣ ξ ∣ d ξ + O ( Cλ ) . We use a Taylor expansion of the exponential: for any r ∈ R , e i rλ = + i rλ + O ( Cr λ ) . Putting this estimate in the left side of (4.23) and using the identity ∣ e ir ∣ = r ∈ R , we obtain ∣∫ [− λπ,λπ ] d [ λ ( e i ξjλ − ) ( e i ξkλ − ) ( e i ξj λ − ) ( e i ξk λ − ) − ξ j ξ k ξ j ξ k ] e iξ ⋅ x e − λ ( ∑ dl = ( − cos ξlλ )) d ξ ∣≤ Cλ ∫ [− λπ,λπ ] d ( ξ j + ξ k + ξ j + ξ k ) e − λ ( ∑ dl = ( − cos ξlλ )) d ξ. Using that there exists a universal constant C such that, for any r ∈ [− π, π ] , 1 − cos ( r ) ≥ r C , we obtain theestimate ∣∫ [− λπ,λπ ] d [ λ ( e i ξjλ − ) ( e i ξkλ − ) ( e i ξj λ − ) ( e i ξk λ − ) − ξ j ξ k ξ j ξ k ] e iξ ⋅ x e − λ ( ∑ di = ( − cos ξiλ )) d ξ ∣ (4.24) ≤ Cλ ∫ [− λπ,λπ ] d ( ξ j + ξ k + ξ j + ξ k ) e − ∣ ξ ∣ C d ξ ≤ Cλ .
A consequence of (4.24) is the identity ∫ [− λπ,λπ ] d λ ( e i ξjλ − ) ( e i ξkλ − ) ( e i ξj λ − ) ( e i ξk λ − ) e iξ ⋅ x e − λ ( ∑ di = ( − cos ξiλ )) d ξ = ∫ [− λπ,λπ ] d ξ j ξ k ξ j ξ k e iξ ⋅ x e − λ ( ∑ di = ( − cos ξiλ )) d ξ + O ( Cλ ) , so that, in order to prove (4.23), it is sufficient to prove(4.25) ∫ [− λπ,λπ ] d ξ j ξ k ξ j ξ k e iξ ⋅ x e − λ ( ∑ di = ( − cos ξiλ )) d ξ = ∫ R d ξ j ξ k ξ j ξ k e iξ ⋅ x e −∣ ξ ∣ d ξ + O ( Cλ ) . . USING THE SYMMETRY AND ROTATION INVARIANCE OF THE DUAL VILLAIN MODEL 169 We now prove (4.25). Using the inequality ∣∫ R d ∖[− λπ,λπ ] d ξ j ξ k ξ j ξ k e iξ ⋅ x e −∣ ξ ∣ d ξ ∣ ≤ e − c ∣ ξ ∣ , we see that to prove (4.25), one needs to show(4.26) λ ∣∫ [− λπ,λπ ] d ξ j ξ k ξ j ξ k e iξ ⋅ x ( e − λ ( ∑ dl = ( − cos ξlλ )) − e −∣ ξ ∣ ) d ξ ∣ ≤ C. The strategy to prove (4.26) is to apply the dominated convergence theorem to prove the (stronger) statement(4.27) λ ∫ [− λπ,λπ ] d ξ j ξ k ξ j ξ k e iξ ⋅ x [ e − λ ( ∑ di = ( − cos ξiλ )) − e −∣ ξ ∣ ] d ξ —→ λ →∞ . To apply the dominated convergence theorem to the left side of (4.27), we verify its assumptions.
Pointwise convergence.
By using the Taylor expansion of the cosine, we have, for any ξ ∈ R d ,2 d ∑ i = ( − cos ξ i λ ) = ∣ ξ ∣ λ + O ( C ∣ ξ ∣ λ ) , which yields(4.28) λ ( d ∑ l = ( − cos ξ l λ )) = ∣ ξ ∣ + O ( C ∣ ξ ∣ λ ) . The expansion (4.28) implies the pointwise convergence, for any ξ ∈ R d , λ ( e − λ ( ∑ di = ( − cos ξiλ )) − e −∣ ξ ∣ ) = e −∣ ξ ∣ λ ( e − λ ( ∑ dl = ( − cos ξlλ )) +∣ ξ ∣ − ) —→ λ →∞ . Uniform upper bound.
We split this argument into two cases.
Case 1. If ∣ ξ ∣ ≤ λ , then the expansion (4.28) implies the bound e −∣ ξ ∣ λ ( e − λ ( ∑ dl = ( − cos ξlλ )) +∣ ξ ∣ − ) ≤ e −∣ ξ ∣ λ ( e C ∣ ξ ∣ λ − ) (4.29) ≤ e −∣ ξ ∣ λe C C ∣ ξ ∣ λ ≤ C ∣ ξ ∣ e −∣ ξ ∣ , where we have used in the second line that for any constant C >
0, the exponential is e C -Lipschitz on theinterval [ , C ] and in the third line the inequality λ ≥ Case 2. If ∣ ξ ∣ ≥ λ , then we use the inequality 1 − cos ( r ) ≥ r C , valid for some universal constant C and forany real number r ∈ [− π, π ] , to obtain ∣ e −∣ ξ ∣ λ ( e − λ ( ∑ dl = ( − cos ξlλ )) +∣ ξ ∣ − )∣ = λ ∣( e − λ ( ∑ dl = ( − cos ξlλ )) − e −∣ ξ ∣ )∣ (4.30) ≤ λe − ∣ ξ ∣ C ≤ C ∣ ξ ∣ e − ∣ ξ ∣ C . A combination of (4.29) and (4.30) implies the upper bound, for any λ ≥ ξ ∈ [− λπ, λπ ] d , ∣ ξ j ξ k ξ j ξ k e iξ ⋅ x λ ( e − λ ( ∑ dl = ( − cos ξlλ )) − e −∣ ξ ∣ )∣ ≤ C ∣ ξ ∣ e − ∣ ξ ∣ C . Since the map ξ ↦ ∣ ξ ∣ e − ∣ ξ ∣ C is integrable over R d , we can apply the dominated convergence theorem andconclude the proof of (4.27). The proof of (4.5) is complete.We use the properties (4.4) and (4.5) to complete the proof of Lemma 4.1. We fix two parameters λ ∈ [ , ∞] and x ∈ R d such that ∣ x ∣ = λx ∈ Z d . We let α ∈ ( , ) be an exponent whose value is decided later in the
70 8. PROOF OF THE ESTIMATES IN CHAPTER 4 argument and shall depend only on the parameters d and ε . We write λ d − F ( λx ) = ( π ) d ∫ ∞ λ d − u ( t, λx ) d t (4.31) = ( π ) d ∫ ∞ λ d + u ( λ t, λx ) d t = ( π ) d ∫ λ − α λ d + u ( λ t, λx ) d t + ( π ) d ∫ ∞ λ − α λ d + u ( λ t, λx ) d t. We estimate the first term in the right side. We use the estimate (4.4) with the exponent k = d + ( π ) d ∫ λ − α λ d + ∣ u ( λ t, λx )∣ d t ≤ C d + ,α ( π ) d λ − α λ d + λ − d − ≤ Cλ − We estimate the second term in the right side of (4.31), we note that since λ is chosen larger than 1, we have λ − α ≥ ( π ) d ∫ ∞ λ − α λ d + u ( λ t, λx ) d t = ( π ) d ∫ ∞ λ − α ∫ R d ξ j ξ k ξ j ξ k e iξ ⋅ x e − t ∣ ξ ∣ d ξ d t + ∫ ∞ λ − α O ( Cλt d + ) d t. Using the estimate ∫ ∞ λ − α t − d + ≤ Cλ α ( d + ) and setting α = εd + , we deduce1 ( π ) d ∫ ∞ λ − α λ d + u ( λ t, λx ) d t = ( π ) d ∫ ∞ λ − α ∫ R d ξ j ξ k ξ j ξ k e iξ ⋅ x e − t ∣ ξ ∣ d ξ d t + O ( Cλ ε − ) . By performing the change of variable ξ ∶= t ξ , we see that ∫ R d ξ j ξ k ξ j ξ k e iξ ⋅ x e − t ∣ ξ ∣ d ξ = t − d + ∫ R d ξ j ξ k ξ j ξ k e iξ ⋅ xt − e − t ∣ ξ ∣ d ξ = t − d + K ( xt ) A combination of the five previous displays shows(4.32) λ d − F ( λx ) = ∫ ∞ λ − α t − d + K ( xt ) + O ( Cλ ε − ) . To complete the argument, note that since the map K belongs to the Schwartz space, it decays faster thanany polynomials at infinity. We thus have the estimate K ( xt ) ≤ Ct d + + α , which implies(4.33) ∫ λ − α t − d + K ( xt ) ≤ C ∫ λ − α t α ≤ C ∫ λ − α λ − ≤ Cλ − . Combining (4.32) and (4.33), we have obtained λ d − F ( λx ) = ∫ ∞ t − d + K ( xt ) + O ( Cλ ε − ) . The proof of Lemma 4.1 is complete. (cid:3)
Lemma 4.1 states that the large-scale behavior of the convolutions of gradients of the discrete Green’sfunction F j,k,j ,k and F j,j is determined by the ( − d ) -homogeneous maps J j,k,j ,k and J j,j . The rest ofthis section is devoted to the study of these maps. One of their properties is that, as ( − d ) -homogeneousfunctions, they belong to the class of tempered distribution. Their Fourier transform turns out to be explicitlycalculable; this is the purpose of the following lemma. Lemma . Fix four integers i, j, k, l ∈ { , . . . , d } . One has the identities ̂ J i,j,k,l ( ξ ) = ξ i ξ j ξ k ξ l ∣ ξ ∣ and ̂ J i,j ( ξ ) = ξ i ξ j ∣ ξ ∣ . Proof.
We only prove the result for the maps J i,j,k,l . Fix a function g ∶ R d → R in the Schwartz spaceand prove the identity ∫ R d J i,j,k,l ( x ) g ( x ) d x = ∫ R d ξ i ξ j ξ k ξ l ∣ ξ ∣ ˆ g ( ξ ) d ξ. . USING THE SYMMETRY AND ROTATION INVARIANCE OF THE DUAL VILLAIN MODEL 171 We first use the definition of the map J i,j,k,l and apply Fubini’s theorem ∫ R d J i,j,k,l ( x ) g ( x ) d x = ∫ R d ∫ ∞ t − d + K i,j,k,l ( xt ) g ( x ) d x = ∫ ∞ t − d + ∫ R d K i,j,k,l ( xt ) g ( x ) d x = ∫ ∞ t − d + ∫ R d t d ˆ K i,j,k,l ( t x ) ˆ g ( ξ ) d ξ. By definition of the map K i,j,k,l , we have the identityˆ K ( ξ ) = ξ i ξ j ξ k ξ l e −∣ ξ ∣ . A combination of the two previous displays shows ∫ R d J i,j,k,l ( x ) g ( x ) d x = ∫ ∞ ∫ R d ξ i ξ j ξ k ξ l e − t ∣ ξ ∣ ˆ g ( ξ ) d ξ = ∫ R d ξ i ξ j ξ k ξ l ∣ ξ ∣ ˆ g ( ξ ) d ξ, as claimed. (cid:3) The next proposition is used in the proof of Theorem 1. It asserts that if a linear combination of the maps F i,j,k,l and F i,j , with a specific structure given by the problem considered in this article, is invariant under thegroup H lattice preserving maps, then it must satisfy the expansion given by (4.35). Proposition . Assume that there exist coefficients ( c ij ) ≤ i,j ≤ d and ( K ij ) ≤ i,j ≤ d , an exponent α > anda map U which is invariant under the group H of the lattice-preserving maps such that (4.34) U ( x ) = d ∑ i,j,k,l = c ij c kl F i,j,k,l ( x ) + d ∑ i,j = K ij F i,j ( x ) + O ( C ∣ x ∣ d − + α ) , then there exists a constant c ∈ R such that (4.35) U ( x ) = c ∣ x ∣ d − + O ( C ∣ x ∣ d − + α ) . Proof.
Applying Lemma 4.1 and Remark 4.2, the expansion (4.34) can be rewritten U ( x ) = d ∑ i,j,k,l = c ij c kl J i,j,k,l ( x ) + d ∑ i,j = K ij J i,j ( x ) + O ( C ∣ x ∣ d − + α ) . Using that the maps J i,j,k,l and J i,j are ( − d ) -homogeneous, we see that the assumption that U is invariantunder the lattice-preserving maps implies that the same property holds for the function ∑ di,j,k,l = c ij c kl J i,j,k,l +∑ di,j = K ij J i,j : for each h ∈ H and each x ∈ Z d ∖ { } , one has d ∑ i,j,k,l = c ij c kl J i,j,k,l ( h ( x )) + d ∑ i,j = K ij J i,j ( h ( x )) = d ∑ i,j,k,l = c ij c kl J i,j,k,l ( x ) + d ∑ i,j = K ij J i,j ( x ) . Using the homogeneity of the maps J i,j,k,l and J i,j , the result can be extended to each point of R d ∖ { } : foreach h ∈ H and each x ∈ R d ∖ { } , one has(4.36) d ∑ i,j,k,l = c ij c kl J i,j,k,l ( h ( x )) + d ∑ i,j = K ij J i,j ( h ( x )) = d ∑ i,j,k,l = c ij c kl J i,j,k,l ( x ) + d ∑ i,j = K ij J i,j ( x ) . To ease the notation, we denote by P the homogeneous polynomial(4.37) P ( ξ ) = ⎛⎝ d ∑ i,j = c ij ξ i ξ j ⎞⎠ + ∣ ξ ∣ d ∑ i,j = K ij ξ i ξ j , so that the Fourier transform of the map ∑ di,j,k,l = c ij c kl J i,j,k,l + ∑ di,j = K ij J i,j is equal to the function ξ ↦ P ( ξ )∣ ξ ∣ by Lemma 4.3.Taking the Fourier transform on both sides of the identity (4.36) and applying Lemma 4.3, we obtain theidentity, for any ξ ∈ R d and any h ∈ H ,(4.38) P ( h ( ξ )) = P ( ξ ) .
72 8. PROOF OF THE ESTIMATES IN CHAPTER 4
We now prove that the equality (4.38) implies that P = a ∣ ξ ∣ . We first use the following fact whose proof isomitted: if a polynomial S ∈ R [ X , . . . , X d ] is homogeneous of degree 4 and is invariant under the lattice-preserving maps, then there exist a, b ∈ R such that S = a d ∑ i = X i + b ∑ ≤ i < j ≤ d X i X j . Applying this result to the polynomial P , we obtain that there exist a, b ∈ R such that, for any ξ ∈ R d ,(4.39) P ( ξ ) = a d ∑ i = ξ i + b ∑ ≤ i < j ≤ d ξ i ξ j . Using the equality ( d ∑ i = ξ i ) = d ∑ i = ξ i + ∑ ≤ i < j ≤ d ξ i ξ j , we can rewrite the identity (4.39) P ( ξ ) = ( a − b ) d ∑ i = ξ i + b ( d ∑ i = ξ i ) . The objective is thus to prove that a − b = P stated in (4.37).We first write(4.40) ⎛⎝ d ∑ i,j = c ij ξ i ξ j ⎞⎠ + ∣ ξ ∣ d ∑ i,j = K ij ξ i ξ j = ( a − b ) d ∑ i = ξ i + b ∣ ξ ∣ , which implies(4.41) ⎛⎝ d ∑ i,j = c ij ξ i ξ j ⎞⎠ + ∣ ξ ∣ ⎛⎝ d ∑ i,j = K ij ξ i ξ j − b ∣ ξ ∣ ⎞⎠ = ( a − b ) d ∑ i = ξ i . To prove that a − b =
0, we argue by contradiction: if a − b ≠
0, then, by (4.40), there exist two real-valuedpolynomials
Q, R homogeneous of degree 2 such that, for any ξ ∈ R d , Q ( ξ ) + ( d ∑ i = ξ i ) R ( ξ ) = d ∑ i = ξ i . We now prove that these two polynomials do not exist. We first reduce the problem to the three dimensionalcase: if we denote by Q , R ∶ R → R the homogeneous polynomials of degree 2 defined by the formulas, foreach ξ = ( ξ , ξ , ξ ) ∈ R , Q ( ξ ) = Q (( ξ , ξ , ξ , , . . . , )) and R ( ξ ) = R (( ξ , ξ , ξ , , . . . , )) , then we have the identity, for each ξ ∈ R ,(4.42) Q ( ξ ) + ( ∑ i = ξ i ) R ( ξ ) = ∑ i = ξ i . It is thus sufficient to prove that the polynomials Q and R do not exist. The proof can be done by an explicitcomputation: if we denote by Q ( ξ ) = Aξ + Bξ + Cξ + Dξ ξ + Eξ ξ + F ξ ξ and R ( ξ ) = Gξ + Hξ + Iξ + Jξ ξ + Kξ ξ + Lξ ξ , expand the left side of (4.42) and identify all the coefficients. We obtain a quadratic system of 15 equationsand 12 variables (the parameters A, B, C, D, E, F, G, H, I, J, K, L ). This system is over-determined; it can besolved explicitly using a software of formal computation and it does not have any solution, ruling out theexistence of the polynomials Q and R . We have thus reached a contradiction and deduce that a − b =
0. Theidentity (4.39) can then be rewritten P ( ξ ) = a ( d ∑ i = ξ i ) = a ∣ ξ ∣ . . TREATING THE ERROR TERM E Q ,Q The equality (4.39) implies that the Fourier transform of the map ∑ di,j,k,l = c ij c kl J i,j,k,l + ∑ di,j = K ij J i,j is equalto a ∣ ξ ∣ , which implies, by taking the inverse Fourier transform, that there exists a constant c such that, for any x ∈ R d ∖ { } ,(4.43) d ∑ i,j,k,l = c ij c kl J i,j,k,l ( x ) + d ∑ i,j = K ij J i,j ( x ) = c ∣ x ∣ d − . Combining the identity (4.43) with the expansion (4.34), we have obtained U ( x ) = c ∣ x ∣ d − + O ( C ∣ x ∣ d − + α ) . The proof of Proposition 4.4 is complete. (cid:3)
5. Treating the error term E q ,q This section is devoted to the treatment the error term E q ,q . It is used in the proof of Theorem 1 inSection 5.2 of Chapter 4. Proposition . Fix two exponents γ, ε ∈ ( , ] such that ε ≤ γ ( d − ) and two charges q , q ∈ Q . Let E q ,q ∶ Z d → R be a function which satisfies the pointwise and L -estimates, for each point κ ∈ Z d and eachradius R ≥ , (5.1) ∣E q ,q ( κ )∣ ≤ C ∣ κ ∣ d − ε and ∑ κ ∈ B R ∖ B R ∣E q ,q ( κ )∣ ≤ CR − γ . Then the constant K q ,q ∶= π ∑ κ ∈ Z d E q ,q ( κ ) is well-defined in the sense that the sum converges absolutelyand one has the expansion (5.2) 4 π ∑ z ,κ ∈ Z d ∇ G ( z ) ⋅ ( n q )∇ G x ( z + κ ) ⋅ ( n q )E q ,q ( κ )= K q ,q ∑ z ∈ Z d ∇ G ( z ) ⋅ ( n q )∇ G ( z − x ) ⋅ ( n q ) + O ⎛⎝ C q ,q ∣ x ∣ d − + γ ( d − ) ⎞⎠ . Proof.
We first write ∇ G ( z ) ⋅ ( n q ) = d ∑ i = ∇ i G ( z )( n q ) i and ∇ G ( z + κ − x ) ⋅ ( n q ) = d ∑ j = ∇ j G ( z + κ − x )( n q ) j and note that to prove the expansion (5.2) it is sufficient to prove that, for each pair of integers i, j ∈ { , . . . , d } ,(5.3) 4 π ∑ z ,κ ∈ Z d ∇ i G ( z )∇ j G x ( z + κ )E q ,q ( κ ) = K q ,q ∑ z ∈ Z d ∇ i G ( z )∇ j G ( z − x ) + O ⎛⎝ C q ,q ∣ x ∣ d − + γ ( d − ) ⎞⎠ . We fix two integers i, j ∈ { , . . . , d } and focus on the proof of (5.3). Using the assumption on the function E q ,q stated in (5.1), one has the estimate ∑ κ ∈ Z d ∣E q ,q ( κ )∣ = ∞ ∑ n = ∑ κ ∈ B n + ∖ B n ∣E q ,q ( κ )∣ ≤ C ∞ ∑ n = − γn < ∞ . This proves that the sum ∑ κ ∈ Z d E q ,q ( κ ) is absolutely convergent and that the constant K q ,q is well-defined.To ease the notation, we let g be the function defined by the formula, for each κ ∈ Z d , g ( κ ) ∶= ∑ z ∈ Z d ∇ i G ( z )∇ j G ( z + κ ) , so that the expansion (5.3) can be rewritten in this notation(5.4) 4 π ∑ κ ∈ Z d g ( κ + x )E q ,q ( κ ) = K q ,q g ( x ) + O ⎛⎝ C q ,q ∣ x ∣ d − + γ ( d − ) ⎞⎠ .
74 8. PROOF OF THE ESTIMATES IN CHAPTER 4
Before proving the expansion (5.4), we record a property of the function g : by using standard estimates on thediscrete Green’s function G , it can be estimated by the formula(5.5) ∣ g ( κ )∣ ≤ ∑ z ∈ Z d ∣∇ i G ( z )∇ j G ( z + κ )∣ ≤ C ∑ z ∈ Z d ∣ z ∣ d − ∣ z + κ ∣ d − ≤ C ∣ κ ∣ d − . The gradient of g can be bounded from above according to the following estimate(5.6) ∣∇ g ( κ )∣ ≤ ∑ z ∈ Z d ∣∇ i G ( z )∇∇ j G ( z + κ )∣ ≤ C ∑ z ∈ Z d ∣ z ∣ d − ∣ z + κ ∣ d ≤ C ln ∣ x ∣∣ κ ∣ d − . To prove the expansion (5.4), it is sufficient to prove the estimate(5.7) ∣ ∑ κ ∈ Z d ( g ( x + κ ) − g ( x )) E q ,q ( κ )∣ ≤ C q ,q ∣ x ∣ d − + γ ( d − ) . To this end, We split the space into three regions.
Region 1. The ball of center and radius ∣ x ∣/ . We use that for each point κ ∈ B ( , ∣ x ∣/ ) , ∣ κ + x ∣ ≥ ∣ x ∣/
2. Thisimplies, by (5.6), the inequality ∣∇ g ( κ )∣ ≤ C ∣ x ∣ d − . From this estimate, we deduce that, for each κ ∈ B ( , ∣ x ∣/ ) , ∣ g ( x + κ ) − g ( x )∣ ≤ ∣ κ ∣ sup z ∈ B ( , ∣ x ∣/ ) ∣∇ g ( x + z )∣ ≤ C ∣ κ ∣ ln ∣ x ∣∣ x ∣ d − . We then use this estimate to compute the sum
RRRRRRRRRRRRRRR ∑ κ ∈ B ( , ∣ x ∣ ) ( g ( x + κ ) − g ( x )) E q ,q ( κ )RRRRRRRRRRRRRRR ≤ C ∑ κ ∈ B ( , ∣ x ∣ ) ∣ κ ∣ ln ∣ x ∣∣ x ∣ d − ∣E q ,q ( κ )∣ . We partition the ball B ( , ∣ x ∣ ) into dyadic annuli according to the inclusion B ( , ∣ x ∣ ) ⊆ ∪ ⌊ ln (∣ x ∣/ )⌋ n = B n + ∖ B n ,where we use the notation ⌊ ln (∣ x ∣/ )⌋ to denote the floor of the real number ln (∣ x ∣/ ) . We additionally notethat for each integer n ∈ N and each point κ ∈ B n + ∖ B n , one has the estimate ∣ κ ∣ ≤ C n . Together with theestimate (5.1) on the error term E q ,q , we obtain RRRRRRRRRRRRRRR ∑ κ ∈ B ( , ∣ x ∣ ) ( G ( x − κ ) − G ( x )) E q ,q ( κ )RRRRRRRRRRRRRRR ≤ C ∑ κ ∈ B ( , ∣ x ∣ ) ln ∣ x ∣ ∣ κ ∣∣ x ∣ d − ∣E q ,q ( κ )∣ (5.8) ≤ C ln ∣ x ∣∣ x ∣ d − ⌊ ln (∣ x ∣/ )⌋ ∑ n = ∣ κ ∣ ∑ κ ∈ B n + ∖ B n ∣E q ,q ( κ )∣≤ C ln ∣ x ∣∣ x ∣ d − ⌊ ln (∣ x ∣/ )⌋ ∑ n = n − γn ≤ C ln ∣ x ∣∣ x ∣ d − ( − γ ) ln (∣ x ∣/ ) ≤ C ∣ x ∣ d − + γ , where we have replaced the exponent γ in the last line by γ to absorb the logarithm. . TREATING THE ERROR TERM E Q ,Q Region 2. The ball of center − x and of radius ∣ x ∣ − γ ( d − ) . In this region, we use the estimate (5.5) on thefunction g and the pointwise bound on the error term E q ,q stated in (5.1) to obtain RRRRRRRRRRRRRRRRRR ∑ κ ∈ B (− x, ∣ x ∣ − γ ( d − ) ) g ( x + κ ) E q ,q ( κ )RRRRRRRRRRRRRRRRRR ≤ C ∑ κ ∈ B ( x, ∣ x ∣ − γ ( d − ) ) ∣ κ − x ∣ d − ∣ κ ∣ d − ε (5.9) ≤ ∣ x ∣ d − ε ∑ κ ∈ B ( x, ∣ x ∣ − γ ( d − ) ) ∣ κ − x ∣ d − ≤ C ∣ x ∣ d − + γ ( d − ) − ε ≤ C ∣ x ∣ d − + γ ( d − ) , where we used in the second inequality the lower bound ∣ κ ∣ ≥ c ∣ x ∣ , valid for any point κ ∈ B ( x, ∣ x ∣ − γ ) , and theassumption ε ≤ γ ( d − ) in the fourth inequality. Region 3. The set C ∶= Z d ∖ ( B ( , ∣ x ∣ ) ∪ B (− x, ∣ x ∣ − γ ( d − ) )) . By the inequality (5.5), we have the estimate,for each κ ∈ C , ∣ g ( x − κ )∣ ≤ C ∣ x ∣ ( − γ ( d − ) )( d − ) . Using the previous estimate and the inclusion C ⊆ ∪ ∞ n =⌈ ln ∣ x ∣/ ⌉ ( B n + ∖ B n ) , one obtains the inequality ∣ ∑ κ ∈ C ( g ( x + κ ) − g ( x )) E q ,q ( κ )∣ ≤ C ∑ κ ∈ C ∣ x ∣ ( − γ ( d − ) )( d − ) ∣E q ,q ( κ )∣ (5.10) ≤ C ∣ x ∣ ( − γ ( d − ) )( d − ) ∞ ∑ n =⌈ ln ∣ x ∣/ ⌉ ∑ κ ∈ B n + ∖ B n ∣E q ,q ( κ )∣≤ C ∣ x ∣ ( − γ ( d − ) )( d − ) ∞ ∑ n =⌈ ln ∣ x ∣/ ⌉ − nγ ≤ C ∣ x ∣ ( − γ ( d − ) )( d − )+ γ . Computing the exponent in the last line of the inequality (5.10) proves the estimate(5.11) ∣ ∑ κ ∈ C ( G ( x − κ ) − G ( x )) E q ,q ( κ )∣ ≤ C ∣ x ∣ ( d − )+ γ . Combining the estimates (5.8), (5.9) and (5.11) completes the proof of Lemma 5.1. (cid:3)
PPENDIX A
Multiscale Poincar´e inequality
Proposition . There exists a constant C ∶= C ( d ) such that for eachcube integer n ∈ N , the following statements hold: (1) For each function f ∈ L (◻ n , µ β ) , ∥ f − ( f ) ◻ n ∥ H − (◻ n ,µ β ) ≤ C ∥ f ∥ L (◻ n ,µ β ) + C n n ∑ m = ∣Z m,n ∣ ⟨( ∣ z + ◻ m ∣ ∑ x ∈ z +◻ m f ( x, ⋅)) ⟩ µ β ;(2) For any function f ∈ L (◻ n , µ β ) , one has the estimate ∥ f − ( f ) ◻ ∥ L (◻ n ,µ β ) ≤ C ∥∇ f ∥ L (◻ n ,µ β ) + C n n ∑ m = ∣Z m,n ∣ ⟨( ∣ z + ◻ m ∣ ∑ x ∈ z +◻ m ∇ f ( x, ⋅)) ⟩ µ β ;(3) for each function f ∈ L (◻ n , µ β ) such that f = on the boundary of the cube ◻ n ∥ f ∥ L (◻ n ,µ β ) ≤ C ∥∇ f ∥ L (◻ n ,µ β ) + C n n ∑ m = ∣Z m,n ∣ ⟨( ∣ z + ◻ m ∣ ∑ x ∈ z +◻ m ∇ f ( x, ⋅)) ⟩ µ β . Proof.
The proof is an almost immediate application of the multiscale Poincar´e inequality proved in [ ,Proposition 1.7 and Lemma 1.8]. We only treat the inequality (1); the other two estimates are similar. Weconsider a field φ ∈ Ω and apply [ , Proposition 1.7 and Lemma 1.8] and a Cauchy-Schwarz inequality to themap x → f ( x, φ ) (with a fixed field φ ). We obtain ∥ f (⋅ , φ ) − ( f (⋅ , φ )) ◻ ∥ H − (◻ n ) ≤ C ∥ f (⋅ , φ )∥ L (◻ n ) + C n n ∑ m = ∣Z m,n ∣ ( ∣ z + ◻ m ∣ ∑ x ∈ z +◻ m f ( x, φ )) . Taking the expectation with respect to the field φ gives ⟨∥ f − ( f ) ◻ ∥ H − (◻ n ) ⟩ µ β ≤ C ∥ f ∥ L (◻ n ) + C n n ∑ m = ∣Z m,n ∣ ⟨( ∣ z + ◻ m ∣ ∑ x ∈ z +◻ m f ( x, ⋅)) ⟩ µ β . We complete the proof by using the estimate ∥ f − ( f ) ◻ n ∥ H − (◻ n ,µ β ) ≤ ⟨∥ f − ( f ) ◻ n ∥ H − (◻ n ) ⟩ µ β , which is a direct consequence of the definitions of the H − (◻) and H − (◻ , µ β ) -norms stated in Chapter 2. (cid:3) PPENDIX B
Solvability of the Neumann problem
In this appendix, we prove the existence and uniqueness of the maximizer in the variational formulation ofthe dual energy ν ∗ used in Chapter 6. We first recall a few definitions.Given a discrete cube ◻ ∶= (− R , R ) d ∩ Z d , we recall the definition of the trimmed cube ◻ − ◻ − ∶= [− R + √ R , R − √ R ] d . We recall the definition of the dual energy E ∗◻ , for each vector p ∗ ∈ R d ×( d ) and each map u ∈ H (◻ , µ β ) , E ∗◻ [ v ] = β ∑ y ∈ Z d ∥ ∂ y v ∥ L (◻ ,µ β ) + ∑ n ≥ ∑ dist ( x,∂ ◻)≥ n β n ∥∇ n + v ( x, ⋅)∥ L ( µ β ) − β ∥∇ v ∥ L (◻∖◻ − ,µ β ) − β ∑ supp q ⊆◻ ⟨∇ q v ⋅ a q ∇ q v ⟩ µ β . The energy E ∗◻ satisfies the coercivity and continuity estimates c (cid:74) v (cid:75) H (◻ ,µ β ) ≤ E ∗◻ [ v ] ≤ C (cid:74) v (cid:75) H (◻ ,µ β ) . The dual subadditive quantity ν ∗ is defined by the formula(0.1) ν ∗ (◻ , p ∗ ) ∶= sup v ∈ H (◻ ,µ β ) − ∣ ◻ ∣ E ∗◻ [ v ] + ∣ ◻ ∣ ∑ x ∈◻ p ∗ ⋅ ⟨∇ v ( x )⟩ µ β . We finally recall that the inverse temperature β is chosen large enough so that all the results of Chapter 5hold with a regularity exponent ε ≪ Proposition . For each p ∗ ∈ R d ×( d ) , there exists a uniquemaximizer of the variational problem (0.1) up to a constant. We denote by v (⋅ , ⋅ , ◻ , p ∗ ) the unique maximizerwhich satisfies ( v (⋅ , ⋅ , ◻ , p ∗ )) ◻ ,µ β = . Additionally, there exists a constant C ∶= C ( d ) > such that it satisfiesthe variance estimate sup x ∈ ◻ var µ β [ v ( x, ⋅ , ◻ , p ∗ )] ≤ C. Proof.
The main difficulty in this proof is the absence of the Poincar´e inequality in both the spatial andfield variables. Indeed if one considers a maximizing sequence ( v n ) n ∈ N in the variational formulation (0.1),then one can prove the upper bounds (cid:74) v n (cid:75) H (◻ ,µ β ) ≤ C. Unfortunately, we do not have a Poincar´e inequality of the form ∥ v n − ( v n ) ◻ ,µ β ∥ L (◻ ,µ β ) ≤ C (cid:74) v n (cid:75) H (◻ ,µ β ) aswe need to integrate over both the field variable φ and the spatial variable x . This implies that we cannotprove boundedness of the L (◻ , µ β ) -norm of the map v n and eventually cannot prove the existence of themaximizer using this technique.To overcome this issue, we introduce a massive term in the variational problem (0.1): for each λ >
0, wedefine(0.2) ν ∗ λ (◻ , p ∗ ) ∶= sup v ∈ H (◻ ,µ β ) − ∣ ◻ ∣ E ∗◻ [ v ] − λ ∥ v ∥ L (◻ ,µ β ) + ∣ ◻ ∣ ∑ x ∈◻ p ∗ ⋅ ⟨∇ v ( x, ⋅)⟩ µ β . In this case, it is clear that the maximizer of the variational problem (0.2) exists and is unique up to a constant.We denote this maximizer by v λ . Additionally, it is clear that one has the estimate, for any λ > ∑ x ∈◻ ⟨ v λ ( x, ⋅)⟩ µ β = (cid:74) v λ (cid:75) H (◻ ,µ ) + λ ∥ v λ ∥ L (◻ ,µ β ) ≤ C. The objective of the argument is to prove an upper bound on the variance of v λ ( x, ⋅) uniform in x ∈ ◻ and λ >
0: we prove that there exists a constant C ∶= C ( d ) such that(0.4) sup λ > ,x ∈ ◻ var µ β [ v λ ( x, ⋅)] ≤ C. The proof can thus be decomposed into two steps: proving the estimate (0.4) and proving that the estimate (0.4)implies Proposition 0.1. We first focus on the second item of the list and show how (0.4) implies the existenceand uniqueness of the maximizer v . We first use the estimates (0.3) and (0.4) to verify that the collection offunctions ( v λ ) λ > is uniformly bounded in the space H (◻ , µ β ) . From the estimate (0.3), we see that we onlyneed to prove the L (◻ , µ β ) -norm estimate(0.5) sup λ > ∥ v λ ∥ L (◻ ,µ β ) < ∞ . To prove the estimate (0.5), we first decompose the L (◻ , µ β ) -norm ∥ v λ ∥ L (◻ ,µ β ) ≤ ∥ v λ − ∣◻∣ ∑ x ∈◻ v λ ( x, ⋅)∥ L (◻ ,µ β ) ·„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„¶ (0.6) −( i ) + ∥ ∣◻∣ ∑ x ∈◻ v λ ( x, ⋅) − v λ ( , ⋅)∥ L (◻ ,µ β ) ·„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„¶ (0.6) −( ii ) (0.6) + ∥ v λ ( , ⋅) − ⟨ v λ ( , ⋅)⟩ µ β ∥ L (◻ ,µ β ) ·„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„¶ (0.6) −( iii ) + ∣⟨ v λ ( , ⋅)⟩ µ β ∣·„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„„„„„„„„„„„„„„¶ (0.6) −( iv ) , and estimate the four terms in the right side separately. For the term (0.6)-(i), we apply the Poincar´e inequalityfor each realization of the field φ . We obtain ∥ v λ − ∣◻∣ ∑ x ∈◻ v λ ( x, ⋅)∥ L (◻ ,µ β ) ≤ CR ∥∇ v λ ∥ L (◻ ,µ β ) ≤ CR.
For the term (0.6)-(ii), we let g be the solution of the discrete Neumann problem ⎧⎪⎪⎪⎨⎪⎪⎪⎩ − ∆ g = δ − ∣◻∣ in ◻ , n ⋅ ∇ g = ∂ ◻ . The solvability is ensured by the identity ∑ x ∈◻ ( δ ( x ) − ∣◻∣ ) =
0. Using the function g , we write ∥ ∣◻∣ ∑ x ∈◻ v λ ( x, ⋅) − v λ ( , ⋅)∥ L (◻ ,µ β ) = ∥ ∑ x ∈◻ ∇ v λ ( x, ⋅) ⋅ ∇ g ( x )∥ L (◻ ,µ β ) ≤ ∥∇ v λ ∥ L (◻ ,µ β ) ∥∇ g ∥ L (◻ ,µ β ) ≤ C ∣◻∣ ∥∇ g ∥ L (◻ ,µ β ) . For the term (0.6)-(iii), we observe that ∥ v λ ( , ⋅) − ⟨ v λ ( , ⋅)⟩ µ β ∥ L (◻ ,µ β ) = var µ β [ v λ ( , ⋅)] ≤ C. For the term (0.6)-(iv), we use identity of (0.3) and the estimate for the term (0.6)-(ii) to write ∣⟨ v λ ( , ⋅)⟩ µ β ∣ = RRRRRRRRRRRR⟨ v λ ( , ⋅)⟩ µ β − ⟨ ∣◻∣ ∑ x ∈◻ v λ ( x, ⋅)⟩ µ β RRRRRRRRRRRR ≤ ∥ ∣◻∣ ∑ x ∈◻ v λ ( x, ⋅) − v λ ( , ⋅)∥ L (◻ ,µ β ) ≤ C ∣◻∣ ∥∇ g ∥ L (◻ ,µ β ) . A combination of the previous displays implies the estimate (0.5). . SOLVABILITY OF THE NEUMANN PROBLEM 181
Since the H (◻ , µ β ) -norm of the family ( v λ ) λ > is bounded uniformly in λ , we can extract a subsequencewhich converges weakly to a map v ∈ H ( µ β ) as λ goes to 0. One can then verify that the map v ∈ H ( µ, β ) isa maximizer in the variational formulation (0.1) and satisfies the estimates(0.7) ( v ) ◻ ,µ β = , (cid:74) v (cid:75) H ( U,µ ) ≤ C and sup x ∈ ◻ var µ β [ v ( x, ⋅)] ≤ C. It only remains to prove (0.4). The argument is similar to the proof of Lemma 3.2 in Chapter 6. We fix a point x ∈ ◻ and apply the Brascamp-Lieb inequality (properly rescaled with respect to the inverse temperature β )to obtain var µ β [ v λ ( x, ⋅)] ≤ Cβ ∑ y,z ∈ Z d ∥ ∂ y v λ ( x, ⋅)∥ L ( µ β ) ∥ ∂ z v λ ( x, ⋅)∥ L ( µ β ) ∣ y − z ∣ d − . Using the definition of v λ as the maximizer of the variational problem (0.2), one sees that it is a solution tothe Hellfer-Sj¨ostrand equation(0.8) { β ∆ φ v λ + β L ◻ v λ + λv λ = ◻ × Ω , n ⋅ ∇ v λ = n ⋅ p ∗ on ∂ ◻ × Ω , where the operator L ◻ is the uniformly elliptic operator defined by the formula L ◻ ∶= − β ∆ + β ∑ n ≥ β n ∇ n + ⋅ ( ◻ ∇ n + ) + β ∇ ⋅ ( ◻∖◻ − ∇) + ∑ supp q ⊆◻ ∇ q ⋅ a q ∇ q . As is mentioned in the proof of Lemma 3.2 of Chapter 6, the operator L ◻ is a perturbation of the Laplacian β ∆. Consequently, the same arguments as the ones developed in Chapter 5 apply and the same regularityresults hold.Applying the operator ∂ to the equation (0.8) as it is done in the proof of Lemma 3.2, we obtain that themap w λ ∶ ( y, z, φ ) → ∂ z v λ ( y, φ ) is a solution of the differentiated Helffer-Sj¨ostrand equation ⎧⎪⎪⎪⎨⎪⎪⎪⎩ β ∆ φ w λ + β L ◻ ,y w λ + β L spat ,z w λ + λw λ = β ∑ supp q ⊆◻ z ( β, q ) sin 2 π ( φ, q ) ( v λ , q ) q y ⊗ q z in ◻ × Z d × Ω , n ⋅ ∇ y w λ = ∂ ◻ × Z d × Ω . The proof is then almost identical to the proof of Lemma 3.2 of Chapter 6; we use the reflection principleto express the function w λ in terms of the Green’s function associated to the differentiated Hellfer-Sj¨ostrandoperator ∆ φ + L ◻ ,y + L spat ,z + λI d . There are two main differences in the argument which are listed below: ● One needs to study the map w λ and not its gradient; this simplifies the computations. ● One needs to study the Green’s function associated to the differentiated operator ∆ φ +L ◻ ,y +L spat ,z + λI d and the weight λI d has to be taken into account. This can be achieved with the same strategy as theone presented in Chapter 5. If we consider a function f ∈ L ( µ β ) , a pair of points ( x , y ) ∈ Z d × Z d and let G λ der , f (⋅ , ⋅ , ⋅ ; x , y ) be the solution of the weighted equation∆ φ G λ der , f + L ◻ ,y G λ der , f + L spat ,z G λ der , f + λβ G λ der , f = f δ , then an application of the Feynman-Kac formula shows the identity G λ der , f ( x, y, φ ; x , y ) = β − ∫ ∞ e − λβ t E φ [ f ( φ t ) P φ ⋅ ◻ ( t, x ; x ) ⊗ P φ ⋅ ( t, y ; y )] dt, where we used the notations introduced at the begining of Chapter 5 and where, given a realizationof the dynamics ( φ t ) t ≥ , the map P φ ⋅ ◻ (⋅ , ⋅ ; y ) is the solution of the parabolic equation ⎧⎪⎪⎨⎪⎪⎩ ∂ t P φ ⋅ ◻ (⋅ , ⋅ ; y ) − L ◻ ,y P φ ⋅ ◻ (⋅ , ⋅ ; y ) = ( , ∞) × Z d ,P φ ⋅ ◻ ( , ⋅ ; y ) = δ on Z d .
82 B. SOLVABILITY OF THE NEUMANN PROBLEM
We then apply the estimate of Proposition 3.1 of Chapter 5, which applies to both functions P φ ⋅ ◻ (⋅ , ⋅ ; y ) and P φ ⋅ (⋅ , ⋅ ; y ) , to obtain the bound ∥G λ der , f ( x, y, ⋅ ; x , y )∥ L p ( µ β ) ≤ Cβ − ∥ f ∥ L p ( µ β ) ∫ ∞ e − λβ t Φ C ( tβ , x − y ) Φ C ( tβ , x − y ) dt (0.9) ≤ Cβ − ∥ f ∥ L p ( µ β ) ∫ ∞ Φ C ( tβ , x − y ) Φ C ( tβ , x − y ) dt ≤ C ∣ x − y ∣ d − + ∣ x − y ∣ d − . In particular, the estimate (0.9) is uniform in the weight λ . (cid:3) PPENDIX C
Basic estimates on discrete convolutions
The objective of this appendix is to prove estimates on some discrete convolutions of functions decayingalgebraically fast at infinity.In the first proposition of this appendix, we consider two exponents α, β > α + β > d and proveestimates on the function y → ∑ x ∈ Z d ∣ x ∣ α ∣ x − y ∣ β . We distinguish different cases depending on the values of the exponents α and β ; the results are collected inthe following proposition. Proposition . Given a pair of exponents α, β > such that α + β > d and a point y ∈ Z d , one has theestimates (i) If α ∈ ( , d ) and β ∈ ( , d ) , ∑ x ∈ Z d ∣ x ∣ α ∣ x − y ∣ β ≤ C ∣ y ∣ α + β − d ;(ii) If α = d and β ∈ ( , d ] , ∑ x ∈ Z d ∣ x ∣ α ∣ x − y ∣ β ≤ C ln ∣ y ∣∣ y ∣ β ;(iii) If α > d and β ∈ ( , ∞) , ∑ x ∈ Z d ∣ x ∣ α ∣ x − y ∣ β ≤ C ∣ y ∣ min ( α,β ) . Proof.
The proof of the points (ii) and (iii) can be found in [ , Appendix]. We only prove (i) anddecompose the space into three regions according to the formula(0.1) ∑ x ∈ Z d ∣ x ∣ α ∣ x − y ∣ β = ∑ x ∈ B (∣ y ∣/ ) ∣ x ∣ α ∣ x − y ∣ β ·„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„¶ (0.1) −( i ) + ∑ x ∈ B ( y, ∣ y ∣/ ) ∣ x ∣ α ∣ x − y ∣ β ·„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„¶ (0.1) −( ii ) + ∑ x ∈ Z d ∖( B (∣ y ∣/ )∪ B ( y, ∣ y ∣/ )) ∣ x ∣ α ∣ x − y ∣ β . ·„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„¶ (0.1) −( iii ) For the term (0.1)-(i), we use the inequality ∣ y − x ∣ ≥ ∣ y ∣ , valid for any point x ∈ B (∣ y ∣ / ) . We obtain ∑ x ∈ B (∣ y ∣/ ) ∣ x ∣ α ∣ x − y ∣ β ≤ C ∣ y ∣ β ∑ x ∈ B (∣ y ∣/ ) ∣ x ∣ α ≤ C ∣ y ∣ α + β − d The term (0.1)-(ii) is estimated similarly, we use this time the inequality ∣ x ∣ ≥ ∣ y ∣ , valid for any point x ∈ B ( y, ∣ y ∣ / ) , and obtain ∑ x ∈ B ( y, ∣ y ∣/ ) ∣ x ∣ α ∣ x − y ∣ β ≤ C ∣ y ∣ β ∑ x ∈ B ( y, ∣ y ∣/ ) ∣ y − x ∣ α ≤ C ∣ y ∣ α + β − d . For the term (0.1)-(iii), we use the inequality ∣ y − x ∣ ≥ c ∣ x ∣ , valid for any point x ∉ B (∣ y ∣ / ) ∪ B ( y, ∣ y ∣ / ) , andobtain ∑ x ∈ Z d ∖( B (∣ y ∣/ )∪ B ( y, ∣ y ∣/ )) ∣ x ∣ α ∣ x − y ∣ β ≤ C ∑ x ∈ Z d ∖( B (∣ y ∣/ )∪ B ( y, ∣ y ∣/ )) ∣ x ∣ α ∣ x ∣ β ≤ C ∑ x ∈ Z d ∖ B (∣ y ∣/ ) ∣ x ∣ α + β ≤ C ∣ y ∣ α + β − d . (cid:3) We record as a corollary three estimates which are used in (5.25) of Chapter 4 and (3.13) of Chapter 6.
Corollary . One has the estimates, for any point x ∈ Z d , (0.2) ∑ z ,z ∈ Z d ∣ x − z ∣ d ∣ z − z ∣ d − ε ∣ z ∣ d − ≤ C ln ∣ x ∣∣ x ∣ d − − ε and (0.3) ∑ z ,z ∈ Z d ∣ x − z ∣ d − × ∣ z − z ∣ d − ε ∣ z ∣ d ≤ C ∣ x ∣ d − − ε . For any pair of points x, x ′ ∈ Z d , one has the estimate (0.4) ∑ y,y ′ ∈ Z d ∣ y ′ − x ′ ∣ d + ∣ y − x ∣ d + ∣ y − y ′ ∣ d − ≤ C ∣ x − x ′ ∣ d − − ε . Proof.
To prove (0.2), we apply Proposition 0.1 twice and obtain ∑ z ,z ∈ Z d ∣ x − z ∣ d ∣ z − z ∣ d − ε ∣ z ∣ d − ≤ ∑ z ∈ Z d ∣ x − z ∣ d ∣ z ∣ d − − ε ≤ C ln ∣ x ∣∣ x ∣ d − − ε . The proof of the estimate (0.3) is similar. We now prove (0.4). By the change of variables y ′ ∶= y ′ − x ′ and y ∶= y − x ′ , one has the identity ∑ y,y ′ ∈ Z d ∣ y ′ − x ′ ∣ d + ∣ y − x ∣ d + ∣ y − y ′ ∣ d − = ∑ y,y ′ ∈ Z d ∣ y ′ ∣ d + ∣ y − x + x ′ ∣ d + ∣ y − y ′ ∣ d − . We apply Proposition 0.1 twice to obtain ∑ y,y ′ ∈ Z d ∣ y ′ ∣ d + ∣ y − x + x ′ ∣ d + ∣ y − y ′ ∣ d − ≤ C ∑ y ′ ∈ Z d ∣ y − x + x ′ ∣ d + ∣ y − y ′ ∣ d − ≤ C ∣ x − x ′ ∣ d − − ε . (cid:3) The following proposition is used in (4.18) of Chapter 7 and (3.18) of Chapter 8.
Proposition . One has the estimates, for each pair of points y, z ∈ Z d , (0.5) ∑ y ∈ Z d ∣ y ∣ d − ∣ y − z ∣ d + − ε + ∣ y + z − y ∣ d + − ε ≤ C ∣ y − z ∣ d + max (∣ y ∣ , ∣ z ∣) d − − ε . and ∑ y ∈ Z d ∣ y ∣ d − ε ∣ y − z ∣ d − ε + ∣ y + z − y ∣ d − ε ≤ C ∣ y − z ∣ d max (∣ y ∣ , ∣ z ∣) d − ε Proof.
We only prove (0.5) and first show the estimate: for each real number a ≥ x ∈ Z d ,(0.6) ∑ y ∈ Z d ∣ y ∣ d − a d + − ε + ∣ x − y ∣ d + − ε ≤ Ca d + max (∣ x ∣ , a ) d − − ε . To prove the inequality (0.6), we distinguish two cases.
Case 1. a ≥ ∣ x ∣ . In that case, there exists a constant c ∶= c ( d ) > a d + − ε + ∣ y − x ∣ d + − ε ≥ c ( a d + − ε + ∣ y ∣ d + − ε ) for any point y ∈ Z d . Using this inequality, we compute ∑ y ∈ Z d ∣ y ∣ d − a d + − ε + ∣ x − y ∣ d + − ε ≤ ∑ y ∈ Z d ∣ y ∣ d − a d + − ε + ∣ x − y ∣ d + − ε ≤ ∑ y ∈ B ( ,a ) ∣ y ∣ d − a d + − ε + ∣ y ∣ d + − ε + ∑ y ∈ Z d ∖ B ( ,a ) ∣ y ∣ d − a d + − ε + ∣ y ∣ d + − ε . We estimate the two terms in the right side separately. For the first term, we write ∑ y ∈ B ( ,a ) ∣ y ∣ d − a d − − ε + ∣ y ∣ d + − ε ≤ ∑ y ∈ B ( ,a ) ∣ y ∣ d − a d + − ε ≤ Ca d − ε . . BASIC ESTIMATES ON DISCRETE CONVOLUTIONS 185 For the second term, we write ∑ y ∈ Z d ∖ B ( ,a ) ∣ y ∣ d − a d + − ε + ∣ y ∣ d + − ε ≤ ∑ y ∈ Z d ∖ B ( ,a ) ∣ y ∣ d − ∣ y ∣ d + − ε ≤ Ca d − ε . A combination of the three previous displays completes the proof of (0.6) in the case a ≥ ∣ x ∣ . Case 2. a ≤ ∣ x ∣ . In that case, an application of Young’s inequality yields the estimate a d + − ε +∣ y − x ∣ d + − ε ≥ ca d + ∣ y − x ∣ d − ε . We deduce that ∑ y ∈ Z d ∣ y ∣ d − a d − − ε + ∣ x − y ∣ d − − ε ≤ C ∑ y ∈ Z d ∣ y ∣ d − a d + ∣ x − y ∣ d − ε . We apply Proposition 0.1 to obtain ∑ y ∈ Z d ∣ y ∣ d − ∣ x − y ∣ d − ε ≤ C ∣ x ∣ d − − ε . A combination of the two previous displays completes the proof of (0.6) in the case a ≤ ∣ x ∣ .We now prove the inequality (0.5). Applying the inequality (0.6) with the choices a = ∣ y − z ∣ and x = y + z ,we obtain ∑ y ∈ Z d ∣ y ∣ d − ∣ y − z ∣ d + − ε + ∣ y + z − y ∣ d + − ε ≤ C ∣ y − z ∣ d + max (∣ y + z ∣ , ∣ y − z ∣) d − − ε . We complete the proof of (0.5) by using the estimate max (∣ y + z ∣ , ∣ y − z ∣) ≥ max (∣ y ∣ , ∣ z ∣) . (cid:3) The next proposition of this appendix is used in (1.15) of Chapter 8.
Proposition . One has the estimate, for any point x ∈ Z d , (0.7) ∑ y,z ∈ Z d ∣ y ∣ d − ∣ x − y ∣ d − ∣ z ∣ d − ∣ z − x ∣ d − ∣ y − z ∣ d − ε ≤ C ∣ x ∣ d − . Proof.
We first fix a point y ∈ Z d and prove the estimate(0.8) ∑ z ∈ Z d ∣ z ∣ d − ∣ z − x ∣ d − ∣ y − z ∣ d − ε ≤ ⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩ C ∣ x ∣ d − ∣ y ∣ d − − ε if ∣ y ∣ ≤ ∣ x ∣ ,C ∣ x ∣ d − ∣ y − x ∣ d − − ε if ∣ y − x ∣ ≤ ∣ x ∣ C ∣ x ∣ d − − ε , otherwise . In the case ∣ y ∣ ≤ ∣ x ∣ , we split the sum according to the formula ∑ z ∈ Z d ∣ z ∣ d − ∣ z − x ∣ d − ∣ y − z ∣ d − ε = ∑ z ∈ B ( , ∣ x ∣/ ) ∣ z ∣ d − ∣ z − x ∣ d − ∣ y − z ∣ d − ε ·„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„¶ (0.9) −( i ) + ∑ z ∈ B ( x, ∣ x ∣/ ) ∣ z ∣ d − ∣ z − x ∣ d − ∣ y − z ∣ d − ε ·„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„¶ (0.9) −( ii ) (0.9) + ∑ z ∈ Z d ∖( B ( , ∣ x ∣/ )∪ B ( x, ∣ x ∣/ )) ∣ z ∣ d − ∣ z − x ∣ d − ∣ y − z ∣ d − ε . ·„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„¶ (0.9) −( iii ) For the term (0.9)-(i), we use that ∣ z − x ∣ ≥ ∣ x ∣ / z ∈ B ( , ∣ x ∣ / ) . We obtain ∑ z ∈ B ( , ∣ x ∣/ ) ∣ z ∣ d − ∣ z − x ∣ d − ∣ y − z ∣ d − ε ≤ ∣ x ∣ d − ∑ z ∈ B ( , ∣ x ∣/ ) ∣ z ∣ d − ∣ y − z ∣ d − ε ≤ ∣ x ∣ d − ∑ z ∈ Z d ∣ z ∣ d − ∣ y − z ∣ d − ε ≤ ∣ x ∣ d − ∣ y ∣ d − − ε , where we used Proposition 0.1 in the third inequality.
86 C. BASIC ESTIMATES ON DISCRETE CONVOLUTIONS
For the term (0.9)-(ii), we use the inequalities ∣ z ∣ ≥ ∣ x ∣ / ∣ z − y ∣ ≥ ∣ x ∣/ z ∈ B ( x, ∣ x ∣/ ) under the assumption ∣ y ∣ ≤ ∣ x ∣ . We obtain ∑ z ∈ B ( x, ∣ x ∣/ ) ∣ z ∣ d − ∣ z − x ∣ d − ∣ y − z ∣ d − ε ≤ C ∣ x ∣ d − ε ∑ z ∈ B ( x, ∣ x ∣/ ) ∣ z − x ∣ d − ≤ C ∣ x ∣ d − − ε . For the term (0.9)-(iii), we use the estimates ∣ z − x ∣ ≥ c ∣ z ∣ and ∣ y − x ∣ ≥ c ∣ z ∣ , valid for any point z ∈ Z d ∖( B (∣ x ∣/ ) ∪ B ( x, ∣ x ∣/ )) under the assumption ∣ y ∣ ≤ ∣ x ∣ . We obtain ∑ z ∈ Z d ∖( B (∣ x ∣/ )∪ B ( x, ∣ x ∣/ )) ∣ z ∣ d − ∣ z − x ∣ d − ∣ y − z ∣ d − ε ≤ ∑ z ∈ Z d ∖ B (∣ x ∣/ )∪ B ( x, ∣ x ∣/ ) ∣ z ∣ d − − ε ≤ ∑ z ∈ Z d ∖ B ( x, ∣ x ∣/ ) ∣ z ∣ d − − ε ≤ C ∣ x ∣ d − − ε . A combination of the four previous displays and the assumption ∣ y ∣ ≤ ∣ x ∣ yields ∑ z ∈ Z d ∣ z ∣ d − ∣ z − x ∣ d − ∣ y − z ∣ d − ε ≤ C ∣ x ∣ d − ∣ y ∣ d − − ε , which completes the proof of (0.8) in the case ∣ y ∣ ≤ ∣ x ∣/
4. The proof of (0.8) in the case ∣ y − x ∣ ≤ ∣ x ∣/ ∣ x ∣ ≤ ∣ x ∣/ z ∶= z − y .There only remains to prove the estimate in the case (0.8) in the third case. We again split the sum intofour terms ∑ z ∈ Z d ∣ z ∣ d − ∣ z − x ∣ d − ∣ y − z ∣ d − ε = ∑ z ∈ B ( , ∣ x ∣/ ) ∣ z ∣ d − ∣ z − x ∣ d − ∣ y − z ∣ d − ε + ∑ z ∈ B ( x, ∣ x ∣/ ) ∣ z ∣ d − ∣ z − x ∣ d − ∣ y − z ∣ d − ε + ∑ z ∈ B ( y, ∣ x ∣/ ) ∣ z ∣ d − ∣ z − x ∣ d − ∣ y − z ∣ d − ε + ∑ z ∈ Z d ∖( B ( , ∣ x ∣/ )∪ B ( x, ∣ x ∣/ )∪ B ( y, ∣ x ∣/ )) ∣ z ∣ d − ∣ z − x ∣ d − ∣ y − z ∣ d − ε . We then estimate the four terms in the right side separately:(i) For the first term, we use the inequalities ∣ y − z ∣ ≥ c ∣ x ∣ and ∣ z − x ∣ ≥ c ∣ x ∣ ;(ii) For the second term, we use the inequalities ∣ z ∣ ≥ c ∣ x ∣ and ∣ y − z ∣ ≥ c ∣ x ∣ ;(iii) For the third term, we use the inequalities ∣ z ∣ ≥ c ∣ x ∣ and ∣ z − x ∣ ≥ c ∣ x ∣ ;(iv) For the fourth term, we use the inequalities ∣ y − z ∣ ≥ c ∣ x ∣ and ∣ z − x ∣ ≥ c ∣ z ∣ .We obtain ∑ z ∈ Z d ∣ z ∣ d − ∣ z − x ∣ d − ∣ y − z ∣ d − ε ≤ C ∣ x ∣ d − − ε ∑ z ∈ B (∣ x ∣/ ) ∣ z ∣ d − + ∣ x ∣ d − − ε ∑ z ∈ B ( x, ∣ x ∣/ ) ∣ z − x ∣ d − + ∣ x ∣ d − ∑ z ∈ B ( y, ∣ x ∣/ ) ∣ y − z ∣ d − ε + ∣ x ∣ d − ε ∑ z ∈ Z d ∖( B (∣ x ∣/ )∪ B ( x, ∣ x ∣/ )∪ B ( y, ∣ x ∣/ )) ∣ z ∣ d − ≤ C ∣ x ∣ d − − ε + ∣ x ∣ d − ε ∑ z ∈ Z d ∖ B (∣ x ∣/ ) ∣ z ∣ d − ≤ C ∣ x ∣ d − − ε . . BASIC ESTIMATES ON DISCRETE CONVOLUTIONS 187 The proof of the estimate (0.8) is complete. We now complete the proof of (0.7). By applying the estimate (0.8),we obtain the inequality ∑ y,z ∈ Z d ∣ y ∣ d − ∣ x − y ∣ d − ∣ z ∣ d − ∣ z − x ∣ d − ∣ y − z ∣ d − ε ≤ C ∣ x ∣ d − ∑ y ∈ B ( , ∣ x ∣/ ) ∣ y ∣ d − − ε ∣ x − y ∣ d − + C ∣ x ∣ d − ∑ y ∈ B ( x, ∣ x ∣/ ) ∣ y ∣ d − ∣ x − y ∣ d − − ε + C ∣ x ∣ d − − ε ∑ y ∈ Z d ∖( B ( , ∣ x ∣/ )∪ B ( x, ∣ x ∣/ )) ∣ y ∣ d − ∣ x − y ∣ d − . We estimate each term on the right side by applying Proposition 0.1. We obtain ∑ y,z ∈ Z d ∣ y ∣ d − ∣ x − y ∣ d − ∣ z ∣ d − ∣ z − x ∣ d − ∣ y − z ∣ d − ε ≤ C ∣ x ∣ d − . The proof of (0.7) is complete. (cid:3)
The following estimate is used in (3.27) and (3.31) of Chapter 6 with the exponent α = d + . Proposition . One has the estimate, for each point y ∈ Z d and each exponent α > d , (0.10) ∑ y ∈ Z d ∣ y ∣ α + ∣ y − y ∣ α ≤ C ∣ y ∣ α − d , where the constant C depends on the parameters α and d . Remark . A variation of the proof gives the following generalization of (0.10): for every cube ◻ ⊆ Z d of center 0 and sidelength R ≥ y ∈ Z d , ∑ y ∈ Z d ∣ y ∣ α + ∣ y − y ∣ α ≤ C max ( R, ∣ y ∣) α − d . Proof.
We split the space into two regions according to the formula ∑ y ∈ Z d ∣ y ∣ α + ∣ y − y ∣ α = ∑ y ∈ B ( , ∣ y ∣/ ) ∣ y ∣ α + ∣ y − y ∣ α + ∑ y ∈ Z d ∖ B ( , ∣ y ∣/ ) ∣ y ∣ α + ∣ y − y ∣ α . We estimate the two terms in the right side separately.For the first term, we use the inequality ∣ y ∣ α + ∣ y − y ∣ α ≥ ∣ y ∣ α α , valid for any point y ∈ B (∣ y ∣ / ) . We obtain ∑ y ∈ B (∣ y ∣/ ) ∣ y ∣ α + ∣ y − y ∣ α ≤ C ∑ y ∈ B (∣ y ∣/ ) ∣ y ∣ α ≤ C ∣ y ∣ α − d . For the second term, we use the inequality ∣ y ∣ α + ∣ y − y ∣ α ≥ ∣ y ∣ α and obtain ∑ y ∈ Z d ∖ B (∣ y ∣/ ) ∣ y ∣ α + ∣ y − y ∣ α ≤ ∑ y ∈ Z d ∖ B (∣ y ∣/ ) ∣ y ∣ α ≤ C ∣ y ∣ α − d . (cid:3) ibliography [1] R. A. Adams and J. J. F. Fournier. Sobolev spaces , volume 140 of
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