aa r X i v : . [ m a t h - ph ] J a n RANDOM-FIELD RANDOM SURFACES
PAUL DARIO, MATAN HAREL, RON PELED
Abstract.
We study how the typical gradient and typical height of a random surface aremodified by the addition of quenched disorder in the form of a random independent externalfield. The results provide quantitative estimates, sharp up to multiplicative constants, in thefollowing cases.It is shown that for real-valued disordered random surfaces of the ∇ φ type with a uniformlyconvex interaction potential: (i) The gradient of the surface delocalizes in dimensions 1 ≤ d ≤ d ≥
3. (ii) The surface delocalizes in dimensions 1 ≤ d ≤ d ≥ integer-valued disordered Gaussian free field: (i) The gradientof the surface delocalizes in dimensions d = , d ≥
3. (ii) Thesurface delocalizes in dimensions d = ,
2. (iii) The surface localizes in dimensions d ≥ d ≥ Introduction
Main results.
The seminal work of Imry and Ma [36] predicted that the addition of aquenched random external field eliminates the magnetization phase transition of low-dimensionalspin systems. This was argued to be a generic phenomenon in two dimensions and to occur alsoin three and four dimensions in systems with continuous symmetry. These predictions wereconfirmed for a broad class of spin systems in the celebrated work of Aizenman and Wehr [3, 4].While the Imry-Ma phenomenon has mostly been studied in the spin system context, it hasbeen recognized that related effects occur also for random surfaces of the ∇ φ type subjectedto quenched disorder in suitable ways [9, 10, 38, 39, 50, 19, 20] (see Section 1.2). In thiswork we study the way in which the fluctuations of random surfaces are enhanced by the ad-dition of quenched randomness in the form of an independent external field, focusing on thelocalization and delocalization behavior of the gradient and heights of real- and integer-valuedsurfaces. Quantitative estimates are obtained in all cases studied, complementing other recentquantitative studies of the Imry–Ma phenomenon [17, 2, 26, 25, 1, 24, 21]. Real-valued random-field random surfaces:
The first class of random surfaces that weconsider are real-valued random surfaces of the ∇ φ type with a uniformly convex interactionpotential, which are subjected to a quenched independent external field in the sense of (1.1)below.We start with a few definitions. Let Z d be the standard d -dimensional integer lattice, inwhich vertices are adjacent if they are equal in all but one coordinate and differ by one in thatcoordinate. For Λ ⊆ Z d , let ∂ Λ be the external vertex boundary of Λ, let E ( Λ ) be the set ofedges of Z d with both endpoints in Λ and set Λ + ∶= Λ ∪ ∂ Λ.Let Λ ⊆ Z d be finite and let η ∶ Λ → R . The Hamiltonian H η Λ of the random surface on Λwith external field η associates to each φ ∶ Λ + → R the energy(1.1) H η Λ ( φ ) ∶= ∑ e ∈ E ( Λ + ) V (∇ φ ( e )) − λ ∑ x ∈ Λ η ( x ) φ ( x ) , where V ∶ R → R is a measurable function satisfying V ( x ) = V (− x ) for all x (the elasticpotential), λ > η and where V (∇ φ ( e )) ∶= V ( φ ( x )− φ ( y )) for an edge e = { x, y } (noting that the orientation of e is immaterial as V is an evenfunction). We assume throughout that the potential V is uniformly convex, i.e., that it is twicecontinuously differentiable and there exist c − , c + satisfying(1.2) 0 < c − ≤ V ′′ ( t ) ≤ c + < ∞ . The probability distribution of the random surface, with zero boundary conditions, is thendefined by(1.3) µ η Λ ( d φ ) ∶= Z η Λ exp (− H η Λ ( φ )) ∏ v ∈ Λ dφ ( v ) ∏ v ∈ ∂ Λ δ ( dφ ( v )) , where dx stands for Lebesgue measure on R , δ is the Dirac delta measure at 0 and where(1.4) Z η Λ ∶= ∫ exp (− H η Λ ( φ )) ∏ v ∈ Λ dφ ( v ) ∏ v ∈ ∂ Λ δ ( dφ ( v )) , the partition function, normalizes µ η Λ to be a probability measure. We denote by ⟨ ⋅ ⟩ µ η Λ theexpectation with respect to µ η Λ , and refer to it as the thermal expectation . We have chosen notto include an inverse temperature parameter multiplying the Hamiltonian in (1.3) as its effectmay be mimicked by multiplying V and λ by β .A natural question pertaining to random surfaces is whether their fluctuations diverge onsequences of domains ( µ Λ n ) n ≥ which increase to Z d . In the absence of an external field (i.e.,when η ≡
0) the following facts are known: In dimensions d = , n tends to infinity [13]; the random surface is delocalized or rough. Indimensions d ≥
3, the Brascamp-Lieb concentration inequality [12, 11] shows that the varianceof the height remains bounded uniformly in n ; the random surface is localized or smooth. TheBrascamp-Lieb inequality further implies that the fluctuations of the (discrete) gradient of thesurface remain bounded in n in every dimension d ≥ η is random, with the random variables ( η ( x )) independentand satisfying various additional assumptions, and we study the properties of µ η Λ for a typicalrealization of η . We shall denote by P the probability measure over the random field, and by E and Var the corresponding expectation and variance.A specific case of interest is the random-field Gaussian free field, i.e., the model (1.3) withthe quadratic potential V ( x ) = x . In this situation, for each realization of the random field η , the random surface has a multivariate Gaussian distribution, and its covariance matrix canbe explicitly calculated. If η is random, independent and each η ( x ) has zero mean and unitvariance, one can prove that, for almost every realization of the random field, the gradient ofthe random surface delocalizes if d ≤ d ≥
3, and that the height of the surfacedelocalizes if d ≤ d ≥
5. The result can be quantified and the typical height ofthe random surface and its gradient can be estimated in every dimension; we refer the readerto [19, Appendix A.1], where the qualitative delocalization of the random-field Gaussian freefield is discussed in dimensions d = ,
4, and to [50, Section 1.2] where the gradient fluctuationsare quantified (see also Section 4.1 for the calculations in the zero-temperature limit). It isshown here that these results extend to the class of potentials satisfying (1.2), for which thelaw of the random surface is not explicitly known.Write Λ L ∶ = { − L, . . . , L } d and let ∣ Λ L ∣ = ( L + ) d be its cardinality. In the next two resultswe consider dimensions d ≥
1, integer L ≥
2, disorder strength λ >
0, ellipticity parameters0 < c − ≤ c + < ∞ and a twice-continuously differentiable V ∶ R → R satisfying V ( x ) = V ( − x ) for all x and the uniform convexity assumption (1.2). We suppose η ∶ Λ L → R are independent ANDOM-FIELD RANDOM SURFACES 3 random variables satisfying additional assumptions as stated in the results and that φ is sampledfrom the measure µ η Λ L given by (1.3). Theorem 1 (Gradient fluctuations, real-valued) . Suppose E [ η ( x )] = and Var [ η ( x )] = forall x ∈ Λ L . There exist C, c > depending only on the dimension d such that the quantity ∥ ∇ φ ∥ L ( Λ + L ,µ η Λ L ) ∶ = ∣ Λ L ∣ ∑ e ∈ E ( Λ + L ) ⟨( ∇ φ ( e )) ⟩ µ η Λ L satisfies d = ∶ cλ c + × L ≤ E [∥ ∇ φ ∥ L ( Λ + L ,µ η Λ L ) ] ≤ Cλ c − × L + Cc − , (1.5) d = ∶ cλ c + × ln L ≤ E [∥ ∇ φ ∥ L ( Λ + L ,µ η Λ L ) ] ≤ Cλ c − × ln L + Cc − , (1.6) d ≥ ∶ cλ c + ≤ E [∥ ∇ φ ∥ L ( Λ + L ,µ η Λ L ) ] ≤ Cλ c − + Cc − . (1.7)We remark that our techniques for controlling the gradient fluctuations are applicable forgeneral external fields η ; see Theorem 6. In addition, our proof of Theorem 1 applies undersignificant relaxations of the uniform convexity assumption (1.2) (in particular, the proof appliesto certain non-convex V ); see Remark 4.2. Theorem 2 (Height fluctuations, real-valued) . Suppose
Var [ η ( x )] = for all x ∈ Λ L . Thereexist C, c > depending only on d and the ratio c + / c − , such that ≤ d ≤ ∶ cλ c + × L − d ≤ Var [⟨ φ ( )⟩ µ η Λ L ] ≤ Cλ c − × L − d , (1.8) d = ∶ cλ c + × ln L ≤ Var [⟨ φ ( )⟩ µ η Λ L ] ≤ Cλ c − × ln L, (1.9) d ≥ ∶ cλ c − ≤ Var [⟨ φ ( )⟩ µ η Λ L ] ≤ Cλ c − . (1.10)Let us make a few remarks about the result. Since the constants in the previous statementsonly depend on the ratio λ / c − , the ellipticity ratio c + / c − and the inverse of c − , one sees that theupper and lower bounds of Theorem 2 hold uniformly over the collection of random variables (⟨ φ ( )⟩ µ ηβ, Λ L ) β ≥ , where the probability measure µ ηβ, Λ L is defined by µ ηβ, Λ L ∶ = Z ηβ, Λ exp ( − βH η Λ ( φ )) ∏ v ∈ Λ dφ ( v ) ∏ v ∈ ∂ Λ δ ( dφ ( v )) . Taking the limit β → ∞ implies that the finite-volume ground configuration of the random-field ∇ φ model (defined in (1.26) below) satisfies the inequalities stated in Theorem 1 andTheorem 2.We point out that the proof of Theorem 2 does not require η to have mean zero (unlike theproof of Theorem 1). It is worth noting, however, that if η is symmetric (i.e., η has the samedistribution as − η ), then E [⟨ φ ( )⟩ µ η Λ L ] = [⟨ φ ( )⟩ µ η Λ L ] = E [⟨ φ ( )⟩ µ η Λ L ] . Thus, a symmetry assumption can be used to upgrade the conclusion of Theorem 2 from avariance bound to a bound on the L -norm (in the random field) of the thermal expectation ⟨ φ ( )⟩ µ η Λ L . It is plausible that such an L bound also holds if the symmetry assumption isweakened to requiring that η has mean zero but this is not proven here; see also Section 7.5. PAUL DARIO, MATAN HAREL, RON PELED
Theorem 2 estimates the extent to which the thermal expectation of the height fluctuatesas the random field changes. It is also natural to consider the full fluctuations of the height,as a result of both thermal fluctuations and the randomness of the field. By the law of totalvariance, this can be decomposed as(1.11) E [⟨ φ ( ) ⟩ µ η Λ L ] − E [⟨ φ ( )⟩ µ η Λ L ] = E [⟨( φ ( ) − ⟨ φ ( )⟩ µ η Λ L ) ⟩ µ η Λ L ] + Var [⟨ φ ( )⟩ µ η Λ L ] with the second term on the right-hand side estimated by Theorem 2. The first term is estimatedby the Brascamp-Lieb inequality [12, 11], which, in our setting, reads(1.12) ⟨( φ ( x ) − ⟨ φ ( x )⟩ µ η Λ L ) ⟩ µ η Λ L ≤ ⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩ CL d = ,C ln L d = ,C d ≥ . We thus see that the first term on the right-hand side of (1.11) is not larger than the secondterm in every dimension (up to a multiplicative factor). We further remark that the Brascamp-Lieb inequality can also be used to obtain Gaussian concentration estimates for the thermalfluctuations (see [23, Section 2.2.1] and [30, Theorem 4.9 and Remark 4.1]).Theorem 1 and Theorem 2 are closely related to the detailed investigations of Cotar andK¨ulske on the existence and uniqueness of gradient Gibbs measures for disordered randomsurfaces [19, 20]. This is discussed further in Section 1.2 but we already point out here that theupper bounds in Theorem 1 in dimensions d ≥ Integer-valued random-field Gaussian free field:
In this section we study the fluc-tuations of the integer-valued
Gaussian free field when subjected to a quenched independentexternal field, as we now define. Let Λ ⊆ Z d be finite and let η ∶ Λ → R . The Hamiltonian H IV ,η Λ of the integer-valued Gaussian free field on Λ with external field η associates to each φ ∶ Λ + → Z the energy H IV ,η Λ ( φ ) ∶ = ∑ e ∈ E ( Λ + ) ∇ φ ( e ) − λ ∑ x ∈ Λ η ( x ) φ ( x ) . where λ > β > µ IV ,β,η Λ ( φ ) ∶ = Z IV ,β,η Λ exp ( − βH IV ,η Λ ( φ )) , to each φ ∶ Λ + → Z satisfying φ ≡ ∂ Λ, where(1.14) Z IV ,β,η Λ ∶ = ∑ φ ∶ Λ + → Z φ ≡ ∂ Λ exp ( − βH IV ,η Λ ( φ )) , the partition function, normalizes µ IV ,β,η Λ to be a probability measure. We denote by ⟨ ⋅ ⟩ µ IV ,β,η Λ the (thermal) expectation with respect to the measure µ IV ,β,η Λ .Our results show that the gradient of the integer-valued random-field Gaussian free fieldshares the delocalization/localization properties of the real-valued surfaces discussed abovein all dimensions, while the two models share similar height fluctuations in dimensions d = ,
2. It is further shown that the integer-valued model localizes in all dimensions d ≥ d = ,
4. The behavior of the integer-valued model in dimensions d ≥ ANDOM-FIELD RANDOM SURFACES 5
In the next results we consider dimensions d ≥
1, integer L ≥
2, inverse temperature β > λ >
0. We suppose η ∶ Λ L → R are independent random variables satisfyingadditional assumptions as stated in the results and that φ is sampled from the measure µ IV ,β,η Λ L given by (1.13). Theorem 3 (Gradient fluctuations, integer-valued) . Suppose E [ η ( x )] = and Var [ η ( x )] = for all x ∈ Λ L . There exist C, c > depending only on d such that the quantity ∥ ∇ φ ∥ L ( Λ + L ,µ IV ,β,η Λ L ) ∶ = ∣ Λ L ∣ ∑ e ∈ E ( Λ + L ) ⟨( ∇ φ ( e )) ⟩ µ IV ,β,η Λ L satisfies d = ∶ cλ × L − C ( + β − ) ≤ E [∥ ∇ φ ∥ L ( Λ + L ,µ IV ,β,η Λ L ) ] ≤ Cλ × L + C ( + β − ) , (1.15) d = ∶ cλ × ln L − C ( + β − ) ≤ E [∥ ∇ φ ∥ L ( Λ + L ,µ IV ,β,η Λ L ) ] ≤ Cλ × ln L + C ( + β − ) , (1.16) d ≥ ∶ cλ − C ( + β − ) ≤ E [∥ ∇ φ ∥ L ( Λ + L ,µ IV ,β,η Λ L ) ] ≤ Cλ + C ( + β − ) . (1.17) Theorem 4 (Height fluctuations, integer-valued, d = , . Assume that η ( x ) has the standardGaussian distribution N ( , ) for all x ∈ Λ L . There exist C, c > such that the quantity ∥ φ ∥ L ( Λ L ,µ IV ,β,η Λ L ) ∶ = ∣ Λ L ∣ ∑ x ∈ Λ L ⟨ φ ( x ) ⟩ µ IV ,β,η Λ L satisfies d = ∶ ce − cλ × L ≤ E [∥ φ ∥ L ( Λ L ,µ IV ,β,η Λ L ) ] ≤ Cλ × L + C ( + β − ) L , (1.18) d = ∶ ce − cλ × L ≤ E [∥ φ ∥ L ( Λ L ,µ IV ,β,η Λ L ) ] ≤ C ( + λ + β − ) × L . (1.19)The theorems determine the order of magnitude of the norms of the gradient and the heightof surface as a function of L , and also estimate the dependence on the disorder strength λ .Comparing to Theorem 1, one notices that the dependence of the gradient norm on the disorderstrength is the same as in the real-valued case. In contrast, compared with Theorem 2, thelower bound at weak disorder for the height norm is significantly smaller (in its dependenceon λ ) in the integer-valued case than in the real-valued case. We expect that the two modelsindeed behave differently. For instance, the proof of Theorem 4 shows that at zero temperature,in order for φ to be non-zero at the origin it is necessary that there exists a connected subsetof Λ L containing the origin in which the sum of the disorder η exceeds its boundary size. Arecent result of Ding and Wirth [24] shows that in two dimensions this is unlikely until L is atleast exp ( λ − / + o ( ) ) .To complete this section, we note that the upper and lower bounds of Theorem 4 are writtenas estimates over the expectation of the spatial L -norm of the field, but a similar, thoughmore involved, argument would yield that these inequalities hold with high probability. Theorem 5 (Height fluctuations, integer-valued, d ≥
3, low temperature and weak disorder) . Suppose d ≥ and assume that η ( x ) has the standard Gaussian distribution N ( , ) for all x ∈ Λ L .There exist β , λ , C, c > such that for all β > β , λ < λ and v ∈ Λ L , it holds that (1.20) P (⟨ { φ ( v ) = } ⟩ µ IV ,β,η Λ L ≤ Ce − β ) ≥ − e cλ . PAUL DARIO, MATAN HAREL, RON PELED
We note that it is possible to take the zero-temperature limit β → ∞ in the results ofTheorem 3, Theorem 4 and Theorem 5 in order to obtain that the ground state of the integer-valued Gaussian free field in the presence of a random field satisfies the estimates stated intheses results.We additionally remark that, while the inequality (1.20) does not formally contradict thedelocalization estimate of Theorem 2, it is possible, though not explicitly written in this article,to combine the arguments of the proof of this result with a Mermin-Wagner type argument(similar to the one presented in [43, Section 1.1]) to obtain the following quantitative delocaliza-tion estimate with high probability in the case of real-valued random surfaces with a Gaussianrandom field: for any R ≥ L ∈ N , P ( − ≤ ⟨ φ ( )⟩ µ η Λ L ≤ ) ≤ ⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩ CL d − d ∈ { , , } ,C √ ln L d = . Background.
Brascamp, Lieb and Lebowitz [13] initiated the first detailed investigationof the fluctuations of real-valued random surfaces of the form (1.3) without an external field(sometimes called the ∇ φ -model) and compared their behavior to the exactly-solvable Gaussianfree field (the case V ( x ) = x ). They proved, among other results, that such surfaces delocalizein two dimensions (the one-dimensional case is classical) and localize in three and higher dimen-sions under the assumption (1.2). The integer-valued Gaussian free field (without an externalfield) behaves similarly in dimensions d = d ≥ roughening transition in two dimensions as the temperature increases: Localization at low temperatures follows bya version of the classical Peierls argument. Delocalization at high temperatures was proved inthe breakthrough work of Fr¨ohlich and Spencer [28, 29] on the Berezinskii-Kosterlitz-Thoulesstransition (see also [37, 40]). Further details on random surfaces without external field can befound, e.g., in the works of Funaki [30], Velenik [51] and Sheffield [48]. The theory of disorderedrandom surfaces is less developed and we survey below some of the existing literature. Real-valued disordered random surfaces:
Van Enter and K¨ulske [50] proved the non-existence of gradient Gibbs measures in dimension d = V is assumed to be even, continuously differentiable and to have super-lineargrowth. They additionally established quantitatively that translation-covariant infinite-volumegradient Gibbs measures have slow decay of correlation when d = V is assumed to have quadratic growth and a second derivative boundedfrom above and when the exterior field is i.i.d. with finite nonzero second moment, and asecond model for which they consider a collection of i.i.d. random functions ( V e ) e ∈ E ( Z d ) withquadratic growth, and study the Gibbs measure formally defined by(1.21) µ ∶ = exp ( − ∑ e V e ( ∇ φ ( e ))) / Z. They prove the existence of the infinite-volume surface tension and of shift-covariant gradientGibbs measures in dimensions d ≥ d ≥ d = ,
2, and that neitherthe infinite-volume surface tension, nor the infinite-volume translation-covariant gradient Gibbsstates can exist if the expectation of the external field is not equal to 0 for the same model.In a second work [20], Cotar and K¨ulske establish the uniqueness of shift-covariant gradientGibbs measures for each fixed expected tilt when the random field is symmetrically distributedin dimensions d ≥ d ≥ ANDOM-FIELD RANDOM SURFACES 7 for the model (1.21). They additionally obtained sharp quantitative decorrelation estimates forgeneral functionals of the gradient of the field for both models.As mentioned above, our Theorem 1 and Theorem 2 are related to the works of Cotar andK¨ulske [19, 20]. It seems that the upper bounds on the gradient fluctuations in Theorem 1in dimensions d ≥ d = ,
2. However, the proofs use different argumentsand make different assumptions on the potential (neither set of assumptions implies the other):The results of [19] are established for potentials V satisfying that V ( t ) ≥ At − B and V ′′ ( t ) ≤ C for some A, B, C > V is only assumed to have super-linear growth. The result is quantifiedand states that, when the domain Λ is a box, the typical height at the center of the box isat least logarithmic in the sidelength of the box. K¨ulske and Orlandi [39] proved that, whenthe potential V is additionally assumed to have a second derivative bounded from above, theheights in the model (1.3) are delocalized (quantitatively) in dimension 2 even in the presenceof an arbitrarily strong δ -pinning. They also obtained an analogous lower bound in higherdimension (exhibiting this time localization depending on the strength of the pinning and ofthe random field), and obtained a lower bound on the fraction of pinned sites in dimensions d ≥ Integer-valued disordered random surfaces:
We are only aware of two earlier studies ofthe integer-valued case, both by Bovier and K¨ulske [9, 10], where they studied the SOS-modelin the presence of a random field whose Hamiltonian is given by the formula H ( φ ) ∶ = ∑ x ∼ y ∣ φ ( x ) − φ ( y )∣ + ε ∑ k ∈ Z ∑ x η x ( k ) { φ ( x ) = k } , where the surface φ is integer-valued and the disorder ( η x ( k )) x ∈ Z d ,k ∈ Z is a collection of i.i.d.Gaussian random variables of variance 1. They proved the existence of infinite-volume Gibbsmeasures in dimensions d ≥ d ≤ Strategy of the arguments.
In this section, we present some of the main argumentsdeveloped in this article. To simplify the presentation, we focus on the setting of real-valuedsurfaces. We present the arguments of the proof of Theorem 1 in Subsection 1.3.1, and theones of the demonstration of Theorem 2 in Subsection 1.3.2. In the case of the integer-valuedGaussian free field, the proofs of Theorem 3 and 4 rely on an extension of the arguments ofSection 1.3.1 to the setting of integer-valued random surfaces. The proof of Theorem 5 is basedon a Peierls-type argument and relies on the results of Fisher–Fr¨ohlich–Spencer [27] (see alsoChalker [16]) and on the bounds on the number of connected sets containing a given vertex ina graph with given maximal degree from Bollob´as [7, Chapter 45].1.3.1.
Gradient fluctuations.
Fix an integer L ≥ η ∶ Λ L → R . Let u Λ L ,η be the zero-temperature random-field random surface for the Gaussian potential V ( x ) = x / λ set to 1; it is defined to be the solution of the discreteelliptic equation (4.1) below. PAUL DARIO, MATAN HAREL, RON PELED
The argument relies on two observations. First, the map u Λ L ,η can be computed in terms ofthe Dirichlet Green’s function (see (4.5)); it depends explicitly on the random field η , and itsvariance can be estimated from above and below. Second, one can prove the two identities, forany point x ∈ Λ L , − ⟨ ∑ e ∋ x V ′ ( ∇ φ ( e ))⟩ µ η Λ L = η ( x ) , (1.22) − ⟨ φ ( x ) ∑ e ∋ x V ′ ( ∇ φ ( e ))⟩ µ η Λ L = + η ( x )⟨ φ ( x )⟩ µ η Λ L , (1.23)where we used the notation ∑ e ∋ x to sum over the directed edges containing x as an endpoint(see (2.2) and (2.4)). We remark that a version of the identity (1.22) was previously usedby van Enter and K¨ulske [50, Proposition 2.2] to establish he non-existence of gradient Gibbsmeasures in dimension d = u Λ L ,η ( x ) , and summing over all the points x ∈ Λ shows the quenchedlower bound(1.24) λc + ∥ ∇ u Λ L ,η ∥ L ( Λ + L ) ≤ ∥ ∇ φ ∥ L ( Λ + L ,µ η Λ L ) , where we used the notation (2.3) to refer to the L -norm of the discrete gradient of the mapping u Λ L ,η . On the other hand, summing the inequality (1.23) over the points x ∈ Λ L , using theuniform convexity of the potential V shows the quenched upper bound(1.25) ∥ ∇ φ ∥ L ( Λ + L ,µ η Λ L ) ≤ c − max {∣ Λ L ∣ , λ ∥ ∇ u Λ L ,η ∥ L ( Λ + ) } . The conclusion of Theorem 1 is then obtained by taking the expectation in the inequalities (1.24)and (1.25) and using the properties of the mapping u Λ L ,η .1.3.2. Height fluctuations.
To highlight the main ideas of the argument of the proof of Theo-rem 2, we only describe the strategy for the upper bounds in the case of the ground state, whenthe field strength λ is set to 1, instead of the thermal expectation of the field under the Gibbsmeasure µ η Λ L . To be more precise, the ground state is defined as the minimizer of the energy J ( v ) ∶ = ∑ e ∈ E ( Λ L ) V ( ∇ v ( e )) − ∑ x ∈ Λ L η ( x ) v ( x ) among all the mappings v ∶ Λ + L → R whose values are set to 0 on the boundary ∂ Λ L . We denoteby v L,η ∶ Λ + L → R the minimizer; it can be equivalently characterized as the unique solution ofthe discrete non-linear elliptic equation(1.26) ⎧⎪⎪⎪⎨⎪⎪⎪⎩ − ∑ e ∋ y V ′ ( ∇ v L,η ( e )) = η ( y ) for y ∈ Λ L ,v L,η ( y ) = y ∈ ∂ Λ L . We wish to estimate the variance (over the random field η ) of the random variable v L,η ( ) ,and prove that it satisfies the same estimates as the ones stated in Theorem 2 for the thermalaverage ⟨ φ ( )⟩ µ η Λ L .The argument relies on the Efron–Stein concentration inequality (stated in Proposition 3.2):if we consider two independent copies of the random field, which we denote by η and ̃ η , andlet η x be the field satisfying η x ( y ) = η ( y ) if y ≠ x and η x ( x ) = ̃ η ( x ) , then we have the varianceestimate(1.27) Var [ v η ( )] ≤ ∑ x ∈ Λ L E [( v L,η ( ) − v L,η x ( )) ] . ANDOM-FIELD RANDOM SURFACES 9
Consequently, it is sufficient, in order to obtain the desired upper bounds, to prove the inequal-ities, for any point x ∈ Λ L ,(1.28) E [( v L,η ( ) − v L,η x ( )) ] ≤ ⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩ CL d = ,C ( ln L ∨ ∣ x ∣ ) d = ,C ∨ ∣ x ∣ d − d ≥ . The proof of the upper bounds (1.28) is based on the observation that the difference w x ∶ = v L,η − v L,η x solves a discrete linear elliptic equation of the form(1.29) { −∇ ⋅ a ∇ w x = ( η ( x ) − ̃ η ( x )) δ x in Λ L ,w x = ∂ Λ L , with δ y the Kronecker delta function, the elliptic operator −∇ ⋅ a ∇ is defined in (2.5), and theenvironment a is explicit, depends on the ground state v η,L and the elastic potential V , andsatisfies the uniform ellipticity estimates c − ≤ a ≤ c + . Using the linearity of the equation (1.29),the mapping w x can be rewritten(1.30) w x ( ) ∶ = ( η ( x ) − ̃ η ( x )) G a ( , x ) . where G a ∶ Λ + L → ( , ∞ ) is the Green’s function associated with the environment a and satisfyingDirichlet boundary condition on the boundary of the box Λ L . Using the uniform ellipticity ofthe environment a , one can obtain upper and lower bounds on the Green’s function G a whichare uniform in the environment a . Using that the random variables η ( x ) and ̃ η x ( x ) have thesame expectation and are of unit variance allows to deduce the upper bound (1.28).The extension of the result from the ground state to thermal expectation over the Gibbsmeasure µ η Λ L as stated in Theorem 2 is done by appealing to the Langevin dynamics associatedwith the models (see Section 3.1), and extending the argument presented above from the settingof elliptic equations to the one of parabolic equations. The details are developed in Section 5.1.4. Organisation of the article.
The rest of the article is organized as follows. Section 2introduces additional notation. Section 3 collects the tools and preliminary results used in theproofs. Sections 4 and 5 treat the case of real-valued random surfaces and are devoted to theproofs of Theorems 1 and 2, respectively. Section 5 is devoted to the integer-valued random-field Gaussian free field and contains the proofs of Theorems 3, 4 and 5. Appendix A provides aproof of the Nash–Aronson estimate used in Section 4, for the heat kernel in a time-dependentuniformly elliptic environment in a box with Dirichlet boundary condition.1.5.
Convention for constants.
Throughout this article, the symbols C and c denote positiveconstants which may vary from line to line, with C increasing and c decreasing. These constantsmay depend only on the dimension d and the ratio of ellipticity c + / c − . Acknowledgements.
We are indebted to David Huse for a discussion on the possible behaviorof the integer-valued random-field Gaussian free field in dimensions d ≥
3, and to MichaelAizenman, Charles M. Newman, Thomas Spencer and Daniel L. Stein for encouragement andhelpful conversations on the topics of this work. We are also grateful to Antonio Auffinger,Wei-Kuo Chen, Izabella Stuhl and Yuri Suhov for the opportunity to present these results inonline talks and for useful discussions. The research of the authors was supported in part byIsrael Science Foundation grants 861/15 and 1971/19 and by the European Research Councilstarting grant 678520 (LocalOrder). Notation
General.
Given a (simple) graph G = ( V ( G ) , E ( G )) we let ⃗ E ( G ) be the set of directededges of G (each edge in E ( G ) appears in ⃗ E ( G ) with both orientations). We write x ∼ y todenote that { x, y } ∈ E ( G ) . We often identify subsets Λ ⊆ G with the induced subgraph of G on Λ. In particular, we write G for V ( G ) and write E ( Λ ) and ⃗ E ( Λ ) for the edges of E ( G ) and ⃗ E ( G ) ), respectively, having both endpoints in Λ. We let ∂ Λ = ∂ G Λ be the external vertexboundary of Λ in G , ∂ Λ ∶ = { x ∈ G ∖ Λ ∶ ∃ y ∈ Λ , y ∼ x } , let Λ + ∶ = Λ ∪ ∂ Λ and let ∣ Λ ∣ be the cardinality of Λ, sometimes referred to as discrete volume .Let a ∧ b be the minimum and a ∨ b be the maximum of a, b ∈ R . Let ⌊ a ⌋ be the floor of a ∈ R .Let Z d be the standard d -dimensional lattice, and let ∣ ⋅ ∣ be the ℓ ∞ -norm on Z d . Twovertices v, w ∈ Z d are adjacent if they are equal in all but one coordinate and differ by one inthat coordinate. Write Λ L ∶ = { − L, . . . , L } d ⊆ Z d for any integer L ≥
0. This is extended to all L ∈ [ , ∞ ) by setting Λ L ∶ = Λ ⌊ L ⌋ .2.2. L -Norms. Let G be a graph. For a function φ ∶ G → R , define the L and normalized L -norms of φ by the formulas(2.1) ∥ φ ∥ L ( G ) ∶ = ( ∑ x ∈ G ∣ φ ( x )∣ ) and ∥ φ ∥ L ( G ) ∶ = ( ∣ G ∣ ∑ x ∈ G ∣ φ ( x )∣ ) . Define the discrete gradient(2.2) ∇ φ ( e ) ∶ = φ ( y ) − φ ( x ) for directed edges e = ( x, y ) ∈ ⃗ E ( G ) . In expressions which do not depend on the orientation of the edge, such as ∣ ∇ φ ( e )∣ or V ( ∇ φ ( e )) ,we allow the edge e to be undirected. For a function v ∶ ⃗ E ( G ) → R satisfying v (( x, y )) = − v (( y, x )) (such as the function ∇ φ ) define the L and normalized L -norms of v by theformulas(2.3) ∥ v ∥ L ( G ) ∶ = ⎛⎝ ∑ e ∈ E ( G ) ∣ v ( e )∣ ⎞⎠ and ∥ v ∥ L ( G ) ∶ = ⎛⎝ ∣ G ∣ ∑ e ∈ E ( G ) ∣ v ( e )∣ ⎞⎠ . Environments and operators.
Introduce the notation, for each vertex x ∈ G ,(2.4) ∑ e ∋ x ∶ = ∑ { e ∈ ⃗ E ( G ) ∶ ∃ y ∈ G,e = ( x,y )} . A map a ∶ E ( G ) → R is called an environment . Its definition is extended to directed edges bysetting a (( x, y )) ∶ = a ({ x, y }) for ( x, y ) ∈ ⃗ E ( Λ ) . The operator −∇ ⋅ a ∇ is defined by the formula(2.5) ∇ ⋅ a ∇ φ ( x ) = ∑ e ∋ x a ( e ) ∇ φ ( e ) for a function φ ∶ G → R and x ∈ G . Unwrapping the definitions shows the following discreteintegration by parts identity, for any pair of functions φ, ψ ∶ G → R ,(2.6) − ∑ x ∈ G ( ∇ ⋅ a ∇ φ ( x )) ψ ( x ) = ∑ e ∈ E ( G ) a ( e ) ∇ φ ( e ) ∇ ψ ( e ) in the sense that if one side converges absolutely then the other converges absolutely to thesame value. Note that the terms inside the sum on the right-hand side are well defined forundirected edges.The above definitions naturally extend to time-dependent environments a ∶ I × E ( G ) → R ,where I ⊆ R is a (time) interval: the operator −∇ ⋅ a ∇ acts on time-dependent functions φ ∶ I × G → R with the same definition (2.5) applied at each fixed time. The identity (2.6) thenholds at each fixed time for time-dependent functions φ, ψ . ANDOM-FIELD RANDOM SURFACES 11
The discrete Laplacian ∆ is the operator ∇ ⋅ a ∇ with a ≡ Tools
In this section, we collect tools pertaining to random surfaces, concentration inequalities andestimates on the solution of parabolic equations which are used in the proofs of Theorem 1 andTheorem 2.3.1.
Langevin dynamics.
The Gibbs measure µ η Λ L (defined in (1.3)) is naturally associatedwith the following dynamics. Definition 3.1 (Langevin dynamics) . Given an integer L ≥
0, random field strength λ , externalfield η ∶ Λ L → R and a collection of independent standard Brownian motions { B t ( x ) ∶ x ∈ Λ L } ,define the Langevin dynamics { φ t ( x ) ∶ x ∈ Λ L } to be the solution of the system of stochasticdifferential equations(3.1) ⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎩ d φ t ( y ) = ⎛⎝ ∑ e ∋ y V ′ ( ∇ φ t ( e )) + λη ( y )⎞⎠ d t + √ B t ( y ) ( t, y ) ∈ ( , ∞ ) × Λ L ,φ t ( y ) = ( t, y ) ∈ ( , ∞ ) × ∂ Λ L ,φ ( y ) = y ∈ Λ L . The Langevin dynamics (3.1) is stationary, reversible and ergodic with respect to the Gibbsmeasure µ η Λ L ; in particular, one has the convergence(3.2) ⟨ φ t ( )⟩ Ð→ t →∞ ⟨ φ ( )⟩ µ η Λ L , where the symbol ⟨ ⋅ ⟩ in the left-hand side refers to the expectation with respect to the Brownianmotions { B t ( x ) ∶ x ∈ Λ L } .3.2. The Efron–Stein inequality.
We record below the Efron–Stein inequality which willbe used in the proof of Theorem 2. A proof can be found in [8, Theorem 3.1].
Proposition 3.2 (Efron–Stein inequality) . Let Λ be a finite set, let η, ̃ η ∶ Λ → R be independentand identically distributed random vectors with independent coordinates, and let f ∶ R Λ → R bea measurable map satisfying E [ f ( η ) ] < ∞ . For each x ∈ Λ set η x to be the field defined by theformula η x ( y ) = η ( y ) if y ≠ x and η x ( x ) = ̃ η ( x ) . Then Var [ f ] ≤ ∑ x ∈ Λ E [( f ( η ) − f ( η x )) ] . Heat kernel bounds.
Let L ≥ < c − ≤ c + < ∞ , let s ∈ R and let y ∈ Λ L . Let a ∶ [ s , ∞ ) × E ( Λ L ) → [ c − , c + ] be a continuous time-dependent (uniformly elliptic)environment. For each initial time s ≥ s , denote by P a = P a ( ⋅ , ⋅ ; s, y ) ∶ [ s, ∞ ) × Λ + L → [ , ] theheat kernel associated with Dirichlet boundary conditions in the box Λ L , i.e., the solution ofthe parabolic equation(3.3) ⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩ ∂ t P a ( t, x ; s, y ) − ∇ ⋅ a ∇ P a ( t, x ; s, y ) = ( t, x ) ∈ ( s, ∞ ) × Λ L ,P a ( t, x ; s, y ) = ( t, x ) ∈ ( s, ∞ ) × ∂ Λ L ,P a ( s, x ; s, y ) = { x = y } x ∈ Λ + L . The maximum principle ensures that P a is indeed non-negative and by summing the equationover x ∈ Λ L it follows that ∑ x ∈ Λ L P a ( t, x ; s, y ) is a non-increasing function of t on [ s, ∞ ) . Inparticular, since the value ∑ x ∈ Λ L P a ( s, x ; s, y ) is equal to 1, one has the estimate, for any t ≥ s ,(3.4) ∑ x ∈ Λ L P a ( t, x ; s, y ) ≤ . Upper and lower bounds on heat kernels are usually referred to as Nash–Aronson estimates.They were initially established by Aronson in [5], in the continuous setting and in infinitevolume for parabolic equations with time-dependent and uniformly elliptic environment. Inthe discrete setting, we refer to the article of Delmotte [22] and the references therein for acollection of heat kernel estimates on general graphs in static environment. The case of discreteparabolic equations with dynamic and uniformly elliptic environment is treated by Giacomin–Olla–Spohn in [33, Appendix B]. The proposition stated below is a finite-volume version oftheir result.
Proposition 3.3 (Nash–Aronson estimates) . In the above setup with L ≥ there exist positiveconstants C , c depending only on the dimension d and the ratio of ellipticity c + / c − such thatthe following holds. For all t ≥ s and x ∈ Λ L , (3.5) P a ( t, x ; s, y ) ≤ C ∨ ( c − ( t − s )) d exp ⎛⎝ − c ∣ x − y ∣ ∨ ( c − ( t − s )) ⎞⎠ exp ( − c c − ( t − s ) L ) . In addition, there exists a constant c > depending only on d and c + / c − such that for any t ≥ s and any ( x, y ) ∈ Λ L × Λ L / satisfying ∣ x − y ∣ ≤ √ c − ( t − s ) ≤ c L , (3.6) P a ( t, x ; s, y ) ≥ c ∨ ( c − ( t − s )) d . The proof of this result is the subject of Appendix A.3.4.
Probability density identities.
Suppose f ∶ R n → [ , ∞ ) is a continuously differentiableprobability density such that ∣ y ∣ f ( y ) tends to zero at infinity and satisfies that y → ( + ∣ y ∣) ∇ f ( y ) is integrable. Integration by parts implies that, for each index 1 ≤ j ≤ n , ∫ R n df ( y ) dy j dy = , (3.7) ∫ R n y j df ( y ) dy j dy = − . (3.8)Let Λ ⊆ Z d be finite and η ∶ Λ → R . Applying the above identities to the probability densityof µ η Λ (see (1.3)) under the assumption (1.2) shows that, for each x ∈ Λ, − ⟨ ∑ e ∋ x V ′ ( ∇ φ ( e ))⟩ µ η Λ = λ η ( x ) , (3.9) − ⟨ φ ( x ) ∑ e ∋ x V ′ ( ∇ φ ( e ))⟩ µ η Λ = + λ η ( x )⟨ φ ( x )⟩ µ η Λ . (3.10) 4. Gradient fluctuations in the real-valued case
The objective of this section is to prove the delocalization of the gradient of the real-valuedrandom-field random surfaces in dimensions d ≤
2, and its localization in dimensions d ≥ The ground state of the random-field Gaussian free field.
Given a finite Λ ⊆ Z d and a function η ∶ Λ → R , let u Λ ,η ∶ Λ + → R be the solution of the Dirichlet problem(4.1) { − ∆ u Λ ,η = η in Λ ,u Λ ,η = ∂ Λ . One readily checks that u Λ ,η is the ground state of the random-field Gaussian free in Λ withzero boundary conditions and unit disorder strength. That is, u Λ ,η minimizes the Hamiltonian H η Λ given by (1.1), with V ( x ) = x and λ =
1, among all functions φ which equal zero on ∂ Λ.The function u Λ ,η will be instrumental in analyzing the gradient fluctuations of the real-valued ANDOM-FIELD RANDOM SURFACES 13 random-field random surfaces and will also play a role in our analyss of the integer-valuedrandom-field Gaussian free field in Section 6.The next proposition studies the order of magnitude of u Λ ,η and its gradient when Λ = Λ L and η is random with independent and normalized values. Proposition 4.1.
Suppose ( η ( x )) x ∈ Λ are independent with zero mean and unit variance. Then(i) The random variables ( u Λ ,η ( x )) x ∈ Λ and ( ∇ u Λ ,η ( e )) e ∈ ⃗ E ( Λ ) have zero mean.Now let Λ = Λ L for an integer L ≥ . There exist constants C, c > depending only on thedimension d such that(ii) The following upper bounds hold for all x ∈ Λ L , (4.2) E [ u Λ L ,η ( x ) ] ≤ ⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩ CL − d ≤ d ≤ ,C ln L d = ,C d ≥ (iii) The following lower bounds hold for all x ∈ Λ cL , (4.3) E [ u Λ L ,η ( x ) ] ≥ ⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩ cL − d ≤ d ≤ ,c ln L d = ,c d ≥ (iv) The averaged L -norm of the gradient of u Λ L ,η satisfies (4.4) E [∥ ∇ u Λ L ,η ∥ L ( Λ + L ) ] ≈ ⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩ L d = , ln L d = , d ≥ , where a ≈ b is used here in the sense c ⋅ a ≤ b ≤ C ⋅ a .Proof. Let G Λ ∶ Λ + × Λ → [ , ∞ ) be the elliptic Green’s function with Dirichlet boundarycondition, defined by requiring that for all y ∈ Λ,(4.5) { − ∆ G Λ ( ⋅ , y ) = δ y in Λ ,G Λ ( ⋅ , y ) = ∂ Λwith δ y the Kronecker delta function. The maximum principle shows that G Λ is indeed non-negative. The linearity of (4.1) and (4.5) implies that(4.6) u Λ ,η ( x ) = ∑ y ∈ Λ G Λ ( x, y ) η ( y ) and ∇ u Λ ,η ( e ) = ∑ y ∈ Λ ∇ G Λ ( e, y ) η ( y ) for x ∈ Λ + and e ∈ ⃗ E ( Λ + L ) . Property (i) is an immediate consequence. In addition, ourassumptions on η imply that(4.7) E [ u Λ ,η ( x ) ] = ∑ y ∈ Λ G Λ ( x, y ) and E [( ∇ u Λ ,η ( e )) ] = ∑ y ∈ Λ ( ∇ G Λ ( e, y )) . Now specialize to the case Λ = Λ L with L ≥
2. Denote by P the solution of the parabolicequation (3.3) in the specific case of the heat equation (i.e., in the case a ≡ G is equal to the integral over the times t ∈ ( , ∞ ) of theheat kernel: we have the identity for any x ∈ Λ + L and y ∈ Λ L , G Λ L ( x, y ) = ∫ ∞ P ( t, x ; 0 , y ) dt. Applying Proposition 3.3 and integrating the inequalities over t ∈ ( , ∞ ) , we obtain the upperbounds(4.8) G Λ L ( x, y ) ≤ ⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩ CL d = ,C ln ( CL ∨ ∣ x − y ∣ ) d = ,C ∨ ∣ x − y ∣ d − d ≥ x, y ∈ Λ L , and the lower bounds(4.9) G Λ L ( x, y ) ≥ ⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎩ cL d = ,c ln ( L ∨ ∣ x − y ∣ ) d = ,c ∨ ∣ x − y ∣ d − d ≥ , valid for y ∈ Λ L / and x ∈ Λ L such that ∣ x − y ∣ ≤ cL , with C, c > d . Combiningthe identity (4.7) with the upper and lower bounds (4.8) and (4.9) shows the upper and lowerbounds on the value of E [ u Λ L ,η ( x ) ] stated in (4.2) and (4.3).We proceed to obtain the estimates on the averaged L -norm of ∇ u Λ L ,η . The identity (4.7)implies that(4.10) E [∥ ∇ u Λ L ,η ∥ L ( Λ + L ) ] = ∣ Λ + L ∣ ∑ e ∈ E ( Λ + L ) , y ∈ Λ L ( ∇ G Λ L ( e, y )) . Applying (2.6) (with G = Λ + L , a ≡ φ = ψ = G Λ L ( ⋅ , y ) ) and using (4.5) we see that(4.11) ∑ e ∈ E ( Λ + L ) ( ∇ G Λ L ( e, y )) = G Λ L ( y, y ) for each y ∈ Λ L . Substituting in (4.10), we obtain E [∥ ∇ u Λ L ,η ∥ L ( Λ + L ) ] = ∣ Λ + L ∣ ∑ y ∈ Λ L G Λ L ( y, y ) . Applying the upper and lower bounds (4.8) and (4.9) with the choice x = y and using thenon-negativity of the Green’s function G Λ L we obtain the inequalities in (4.4). (cid:3) Gradient fluctuations.
In this section, we prove Theorem 1 for the uniformly convex ∇ φ -model, as a consequence of the following result which holds for each deterministic η . Theorem 1follows from the result upon setting Λ = Λ L and letting η be random as in the theorem andthen taking an expectation over η and applying the bounds in item (iv) of Proposition 4.1. Theorem 6.
Let Λ ⊆ Z d be finite, λ > and < c − ≤ c + < ∞ . Let η ∶ Λ → R and let u Λ ,η ∶ Λ + → R be the solution of (4.1) . Suppose φ is sampled from the measure µ η Λ of (1.3) , where the potential V satisfies (1.2) . Then the quantity (4.12) ∥ ∇ φ ∥ L ( Λ + ,µ η Λ ) ∶ = ⎛⎝ ∑ e ∈ E ( Λ + ) ⟨( ∇ φ ( e )) ⟩ µ η Λ ⎞⎠ satisfies (4.13) λc + ∥ ∇ u Λ ,η ∥ L ( Λ + ) ≤ ∥ ∇ φ ∥ L ( Λ + ,µ η Λ ) ≤ λc − ∥ ∇ u Λ ,η ∥ L ( Λ + ) + √ ∣ Λ ∣ c − . ANDOM-FIELD RANDOM SURFACES 15
Proof.
We start with the proof of the lower bound. Multiply the identity (3.9) by u Λ ,η ( x ) andsum over all x ∈ Λ to obtain(4.14) − ∑ x ∈ Λ u Λ ,η ( x )⟨ ∑ e ∋ x V ′ ( ∇ φ ( e ))⟩ µ η Λ = λ ∑ x ∈ Λ u Λ ,η ( x ) η ( x ) . The left-hand side is developed by observing that each edge appears in the sum there with bothorientations and that V ′ is an odd function, − ∑ x ∈ Λ u Λ ,η ( x )⟨ ∑ e ∋ x V ′ ( ∇ φ ( e ))⟩ µ η Λ = ∑ e ∈ E ( Λ + ) ∇ u Λ ,η ( e )⟨ V ′ ( ∇ φ ( e ))⟩ µ η Λ (where the boundary terms are added by noting that u Λ ,η ≡ ∂ Λ). The right-hand sideof (4.14) is developed as ∑ x ∈ Λ u Λ ,η ( x ) η ( x ) = − ∑ x ∈ Λ u Λ ,η ( x ) ∆ u Λ ,η ( x ) = ∑ e ∈ E ( Λ + ) ( ∇ u Λ ,η ( e )) = ∥ ∇ u Λ ,η ∥ L ( Λ + ) by using the equation (4.1) and the equality (2.6). Substituting the last two equalities in (4.14)shows that(4.15) λ ∥ ∇ u Λ ,η ∥ L ( Λ + ) = ∑ e ∈ E ( Λ + ) ∇ u Λ ,η ( e )⟨ V ′ ( ∇ φ ( e ))⟩ µ η Λ . An application of the Cauchy-Schwarz inequality to the right-hand side then gives(4.16) λ ∥ ∇ u Λ ,η ∥ L ( Λ + ) ≤ ∥⟨ V ′ ( ∇ φ )⟩ µ η Λ ∥ L ( Λ + ) . Finally, as V is an even function satisfying (1.2), it follows that(4.17) c − x ≤ V ′ ( x ) x ≤ c + x for all x ∈ R . In particular, ∣ V ′ ( x )∣ ≤ c + ∣ x ∣ . Substituting in (4.16) shows that λ ∥ ∇ u Λ ,η ∥ L ( Λ + ) ≤ c + ∥⟨∣ ∇ φ ∣⟩ µ η Λ ∥ L ( Λ + ) . The Cauchy-Schwarz inequality shows that ⟨∣ ∇ φ ( e )∣⟩ µ η Λ ≤ ⟨( ∇ φ ( e )) ⟩ µ η Λ and the lower bound ofthe theorem follows.We proceed to prove the upper bound. Sum the identity (3.10) over all x ∈ Λ to obtain(4.18) − ∑ x ∈ Λ ⟨ φ ( x ) ∑ e ∋ x V ′ ( ∇ φ ( e ))⟩ µ η Λ = ∣ Λ ∣ + λ ∑ x ∈ Λ + η ( x )⟨ φ ( x )⟩ µ η Λ (where the terms with x ∈ ∂ Λ are added by noting that φ ≡ ∂ Λ). As for the lower bound,the left-hand side is developed by summing over the two orientations of each edge, − ∑ x ∈ Λ ⟨ φ ( x ) ∑ e ∋ x V ′ ( ∇ φ ( e ))⟩ µ η Λ = ∑ e ∈ E ( Λ + ) ⟨ ∇ φ ( e ) V ′ ( ∇ φ ( e ))⟩ µ η Λ . The right-hand side of (4.18) is developed as ∑ x ∈ Λ η ( x )⟨ φ ( x )⟩ µ η Λ = − ∑ x ∈ Λ + ∆ u Λ ,η ( x )⟨ φ ( x )⟩ µ η Λ = ∑ e ∈ E ( Λ + ) ∇ u Λ ,η ( e )⟨ ∇ φ ( e )⟩ µ η Λ by using the equation (4.1) and the equality (2.6). Substituting in (4.18) shows that(4.19) ∑ e ∈ E ( Λ + ) ⟨ ∇ φ ( e ) V ′ ( ∇ φ ( e ))⟩ µ η Λ = ∣ Λ ∣ + λ ∑ e ∈ E ( Λ + ) ∇ u Λ ,η ( e )⟨ ∇ φ ( e )⟩ µ η Λ . The left-hand side of the equality is developed using the lower bound in (4.17) while the right-hand side is developed using the Cauchy-Schwarz inequality, yielding(4.20) c − ∑ e ∈ E ( Λ + ) ⟨( ∇ φ ( e )) ⟩ µ η Λ ≤ ∣ Λ ∣ + λ ∥ ∇ u Λ ,η ∥ L ( Λ + ) ∥⟨ ∇ φ ⟩ µ η Λ ∥ L ( Λ + ) . Recalling the definition of ∥ ∇ φ ∥ L ( Λ + ,µ η Λ ) from (4.12) and using that a + b ≤ ( a, b ) for a, b > ⟨ ∇ φ ( e )⟩ µ η Λ ≤ ⟨ ∇ φ ( e ) ⟩ µ η Λ we conclude that(4.21) c − ∥ ∇ φ ∥ L ( Λ + ,µ η Λ ) ≤ {∣ Λ ∣ , λ ∥ ∇ u Λ ,η ∥ L ( Λ + ) ∥ ∇ φ ∥ L ( Λ + ,µ η Λ ) } . Rearranging we get(4.22) ∥ ∇ φ ∥ L ( Λ + ,µ η Λ ) ≤ max ⎧⎪⎪⎨⎪⎪⎩√ ∣ Λ ∣ c − , λc − ∥ ∇ u Λ ,η ∥ L ( Λ + ) ⎫⎪⎪⎬⎪⎪⎭ which implies the upper bound of the theorem. (cid:3) Remark 4.2.
An inspection of the proof of Theorem 6 shows that the assumption that V satisfies (1.2) may be relaxed. Specifically, the proof of the lower bound in Theorem 6 requiresonly that V is sufficiently smooth and has sufficient growth at infinity for the identity (3.9)to hold and that ∣ V ′ ( x )∣ ≤ c + ∣ x ∣ for all x ∈ R . Similarly, the proof of the upper bound inTheorem 6 requires only that V is sufficiently smooth and has sufficient growth at infinity forthe identity (3.10) to hold and that V ′ ( x ) x ≥ c − x for all x ∈ R . Note that V need not beconvex for these relaxed assumptions to hold.Consequently, the lower and upper bounds in Theorem 1 hold under the same relaxed as-sumptions. 5. Height fluctuations in the real-valued case
In this section, we study the fluctuations of the height of real-valued random-field randomsurfaces. We prove that the surface delocalizes in dimensions 1 ≤ d ≤ d ≥
5. proving Theorem 2. Quantitative upper and lower bounds for the varianceof the thermal expectation of the height are obtained. The section is organized as follows.In Subsection 5.1, we establish a quantitative theorem which estimates the difference of thethermal expectations of the height of the random surface at the center of a box with twodifferent external fields. Subsection 5.2 is devoted to the proof of the upper bounds of theinequalities (1.5), (1.6) and (1.7) by combining the results of Subsection 5.1 with the Efron–Stein inequality following the outline presented in Section 1.3. In Section 5.3, we prove thelower bound of the inequalities (1.5), (1.6) and (1.7), and thus complete the proof of Theorem 2.5.1.
A quantitative estimate for the thermal expectation of the height of the randomsurface.
This section is devoted to the proof of an upper bound and a lower bound on thedifference of the thermal expectations of the height at the center of a box with two different(and arbitrary) external fields. The argument relies on a coupling argument for the Langevindynamics associated with the random-field ∇ φ -model and on the Nash–Aronson estimate forparabolic equation with uniformly convex and time-dependent environment (Proposition 3.3). Theorem 7.
Let L ≥ , λ > and η, η ∶ Λ L → R be two external fields. Then there exists aconstant C ∈ ( , ∞ ) depending only on d and the ratio of ellipticity c + / c − such that (5.1) ∣⟨ φ ( )⟩ µ η Λ L − ⟨ φ ( )⟩ µ η Λ L ∣ ≤ ⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩ λCc − L ∑ x ∈ Λ L ∣ η ( x ) − η ( x )∣ d = ,λCc − ∑ x ∈ Λ L ln ( CL ∨ ∣ x ∣ ) ∣ η ( x ) − η ( x )∣ d = ,λCc − ∑ x ∈ Λ L ∣ η ( x ) − η ( x )∣ ∨ ∣ x ∣ d − d ≥ . ANDOM-FIELD RANDOM SURFACES 17
In addition, there exist two constants c, ̃ c ∈ ( , ∞ ) depending on d and c + / c − such that, if thereexists a vertex x ∈ Λ ̃ cL such that η = η on Λ L ∖ { x } and η ( x ) ≠ η ( x ) , then (5.2) ⟨ φ ( )⟩ µ η Λ L − ⟨ φ ( )⟩ µ η Λ L η ( x ) − η ( x ) ≥ ⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩ λcc − L d = ,λcc − ln ( L ∨ ∣ x ∣ ) d = ,λcc − ∨ ∣ x ∣ d − d ≥ . The following proof uses the idea of coupling two different Langevin dynamics with the sameBrownian motion. This argument was originally used by Funaki and Spohn in [31].
Proof.
Let us fix an integer L ≥
2, two external fields η, η ∶ Λ L → R , consider a collection ofindependent Brownian motions { B t ( x ) ∶ x ∈ Λ L } , and run the two Langevin dynamics (withthe same Brownian motions)(5.3) ⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩ d φ t ( y ) = ∑ e ∋ y V ′ ( ∇ φ t ( e )) d t + λη ( y ) d t + √ B t ( y ) ( t, y ) ∈ ( , ∞ ) × Λ L ,φ ( y ) = y ∈ Λ L ,φ t ( y ) = ( t, y ) ∈ ( , ∞ ) × ∂ Λ L , and(5.4) ⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩ d φ t ( y ) = ∑ e ∋ y V ′ ( ∇ φ t ( e )) d t + λη ( y ) d t + √ B t ( y ) ( t, y ) ∈ ( , ∞ ) × Λ L ,φ ( y ) = y ∈ Λ L ,φ t ( y ) = ( t, y ) ∈ ( , ∞ ) × ∂ Λ L . By ergodicity of the Langevin dynamics stated in Subsection 3.1, we have the convergences(5.5) ⟨ φ t ( )⟩ Ð→ t →∞ ⟨ φ ( )⟩ µ η Λ L and ⟨ φ t ( )⟩ Ð→ t →∞ ⟨ φ ( )⟩ µ η Λ L . Taking the difference between the two stochastic differential equations (5.3) and (5.4), andusing that the two driving Brownian motions are the same, we obtain(5.6) ⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩ ∂ t ( φ t ( y ) − φ t ( y )) = ∑ e ∋ y ( V ′ ( ∇ φ t ( e )) − V ′ ( ∇ φ t ( e ))) − λ ( η ( y ) − η ( y )) ( t, y ) ∈ ( , ∞ ) × Λ L ,φ ( y ) − φ ( y ) = y ∈ Λ L ,φ t ( y ) − φ t ( y ) = ( t, y ) ∈ ( , ∞ ) × ∂ Λ L . The strategy is then to rewrite the equation (5.6) as a discrete linear parabolic equation. Tothis end, we note that, since the potential V is assumed to be twice continuously differentiable,the following identity is valid, for any edge e ∈ E ( Λ L ) and any time t ≥ V ′ ( ∇ φ t ( e )) − V ′ ( ∇ φ t ( e )) = ( ∫ V ′′ ( s ∇ φ t ( e ) + ( − s ) ∇ φ t ( e )) d s ) × ( ∇ φ t ( e ) − ∇ φ t ( e )) . Thus, if we let a ∶ [ , ∞ ) × E ( Λ + L ) → [ , ∞ ) be the time-dependent environment defined by theformula(5.7) a ( t, e ) ∶ = ∫ V ′′ ( s ∇ φ t ( e ) + ( − s ) ∇ φ t ( e )) d s, then we have the identity, for any point y ∈ Λ L , ∑ e ∋ y ( V ′ ( ∇ φ t ( e )) − V ′ ( ∇ φ t ( e ))) = ∇ ⋅ a ∇ ( φ t − φ t ) ( y ) , where we used the notation (2.5) for discrete elliptic operators. The inequalities c − ≤ V ′′ ≤ c + imply that the environment a is uniformly elliptic and satisfies c − ≤ a ( t, y ) ≤ c + for any pair ( t, e ) ∈ [ , ∞ ) × E ( Λ L ) . Denoting by w t ( y ) ∶ = φ t ( y ) − φ t ( y ) , we obtain that the map w solvesthe discrete parabolic equation(5.8) ⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩ ∂ t w t ( y ) = ∇ ⋅ a ∇ w t ( y ) + ( η ( y ) − η ( y )) ( t, y ) ∈ ( , ∞ ) × Λ L ,w t ( y ) = ( t, y ) ∈ ( , ∞ ) × ∂ Λ L ,w ( y ) = y ∈ Λ L . Using the notations introduced in Section 3.3, we denote by P a the heat kernel associated withthe environment a . The Duhamel’s formula applied to the equation (5.8) yields the identity(5.9) w t ( ) ∶ = λ ∑ x ∈ Λ L ( η ( x ) − η ( x )) ∫ t P a ( t, s, x ) ds. By the Nash–Aronson estimate on the heat-kernel P a stated in Proposition 3.3, we obtain theinequality, for any time t ≥ ∫ t P a ( t, s, x ) ds ≤ ∫ t C ∨ ( c − s ) d exp ⎛⎝ − c ∣ x − y ∣ ∨ ( c − s ) ⎞⎠ exp ( − c c − sL ) ds ≤ ⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩ CLc − d = ,Cc − ln ( CL ∨ ∣ x ∣ ) d = ,Cc − ∨ ∣ x ∣ d − d ≥ . Taking the expectation with respect to the collection of Brownian motions { B t ( y ) ∶ y ∈ Λ L } and using the identities (5.5), we obtain ∣⟨ φ ( )⟩ µ η Λ L − ⟨ φ ( )⟩ µ η Λ L ∣ ≤ lim sup t →∞ ∣⟨ φ t ( )⟩ − ⟨ φ t ( )⟩∣ (5.10) ≤ lim sup t →∞ ∣⟨ w ( t, )⟩∣ ≤ ⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩ λCc − L ∑ x ∈ Λ L ∣ η ( x ) − η ( x )∣ d = ,λCc − ∑ x ∈ Λ L ln ( CL ∨ ∣ x ∣ ) ∣ η ( x ) − η ( x )∣ d = ,λCc − ∑ x ∈ Λ L ∣ η ( x ) − η ( x )∣ ∨ ∣ x ∣ d − d ≥ . This is the inequality (5.1). We now focus on the proof of (5.2). Since the two external fields η and η are equal everywhere except at the vertex x , the identity (5.9) becomes w t ( ) ∶ = λ ( η ( x ) − η ( x )) ∫ t P a ( t, s, x ) ds. Let us denote by c the constant which appears in the statement of Proposition 3.3 and set ̃ c ∶ = c /
2. Assuming that the vertex x belongs to the box Λ ̃ cL , we may apply the lower boundfor the map P a stated in (3.6) and use that the heat kernel is non-negative to obtain, for any ANDOM-FIELD RANDOM SURFACES 19 time t ≥ c L / c − , ∫ t P a ( t, s, x ) ds ≥ ∫ t −∣ x ∣ / c − t − c L / c − P a ( t, s, x ) ds ≥ ∫ t −∣ x ∣ / c − t − c L / c − c ( c − ( t − s )) d ∨ ds ≥ c − ∫ c L ∣ x ∣ cs d ∨ ds. The term in the right-hand side can be explicitly computed, and we obtain the lower bound,for any time t ≥ c L / c − ,(5.11) ∫ t P a ( t, s, x ) ds ≥ ⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩ cc − L d = ,cc − ln ( L ∨ ∣ x ∣ ) d = ,cc − ∨ ∣ x ∣ d − d ≥ . A similar computation as in (5.10) completes the proof of the lower bound (5.2). (cid:3)
Real-valued random-field random surface models: upper bounds.
In this section,we establish the upper bounds of the inequalities (1.5), (1.6) and (1.7) by combining the resultof Theorem 7 with the Efron–Stein concentration inequality stated in Proposition 3.2.
Proof of Theorem 2: Upper bounds of (1.5) , (1.6) and (1.7) . Let us fix an integer L ≥
2. Thestrategy is to apply the Efron–Stein inequality with the map f ∶ η ↦ ⟨ φ ( )⟩ µ η Λ L . Let us notethat, by the inequality (5.1) applied with η = ⟨ φ ( )⟩ µ L =
0, the L -normof the mapping η ↦ ⟨ φ ( )⟩ µ η Λ L is finite; the assumption of Proposition 3.2 is thus satisfied.We let η and ̃ η be two independent copies of the random field. For each point x ∈ Λ L , wedenote by η x the resampled random field defined by the formula η x ( y ) = η ( y ) if y ≠ x and η x ( y ) = ̃ η ( x ) . By Proposition 3.2, one has the estimate(5.12) Var [⟨ φ ( )⟩ µ η Λ L ] ≤ ∑ x ∈ Λ L E [(⟨ φ ( )⟩ µ η Λ L − ⟨ φ ( )⟩ µ ηx Λ L ) ] . Applying the quenched upper bound (5.1) obtained in Theorem 7 with the external fields η ∶ = η and η ∶ = η x , we obtain the upper bound (⟨ φ ( )⟩ µ η Λ L − ⟨ φ ( )⟩ µ ηx Λ L ) ≤ ⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩ λ Cc − L ( η ( x ) − ̃ η ( x )) d = ,λ Cc − ( ln L ∨ ∣ x ∣ ) ( η ( x ) − ̃ η ( x )) d = ,λ Cc − ( η ( x ) − ̃ η ( x )) ∨ ∣ x − y ∣ d − d ≥ . Taking the expectation in the previous display (with respect to the random field η ) and usingthat the random variables η ( x ) and ̃ η ( x ) have expectation 0 and variance 1 shows the inequality, for any x ∈ Λ L ,(5.13) E [(⟨ φ ( )⟩ µ η Λ L − ⟨ φ ( )⟩ µ ηx Λ L ) ] ≤ ⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩ λ Cc − L d = ,λ Cc − ( ln CL ∨ ∣ x ∣ ) d = ,λ Cc − ∨ ∣ x ∣ d − d ≥ . Summing the inequality (5.13) over all the points x ∈ Λ L and using the estimate (5.12) completesthe proof of the upper bounds of (1.8), (1.9) and (1.10). (cid:3) Real-valued random-field random surface models: lower bounds.
The objectiveof this section is to prove the lower bounds for the variance (in the random field) of the thermalaverage of the heights.
Proof of Theorem 2: lower bounds of (1.8) , (1.9) and (1.10) . We fix an integer L ≥ ̃ c be the constant which appears in the statement of Theorem 7. We let x , . . . x ( L + ) d bean enumeration of the points of the box Λ L such that the collection x , . . . , x ( ⌊̃ cL ⌋+ ) d is anenumeration of the vertices of the box Λ ̃ cL . We let η and ̃ η be two independent copies of therandom field. For each integer k ∈ { , . . . , ( L + ) d } , we introduce the notations η k ∶ = η ( x k ) , ̃ η k ∶ = ̃ η ( x k ) , denote by ν k the law of η k , and let F k be the sigma-algebra generated by the randomvariables η , . . . , η k . We additionally denote by η k the random field satisfying η k ( x ) = η ( x ) if x ≠ x k , and η k ( x k ) = ̃ η ( x k ) .For each integer k ∈ { , . . . , ( L + ) d } , we introduce two random variables. The first one isa function of the values of the field η , . . . , η k and is given by the formula(5.14) X k = X k ( η , . . . , η k ) = E [⟨ φ ( )⟩ µ η Λ L ∣ F k ] ( η , . . . , η k ) = ∫ ⟨ φ ( )⟩ µ η Λ L ∏ j > k ν j ( dη j ) . The second one is a function of the values of the field η , . . . , η k − , ̃ η k and is given by the formula(5.15) ̃ X k = ̃ X k ( η , . . . , η k − , ̃ η k ) = X k ( η , . . . , η k − , ̃ η k ) . Using that the random variable ̃ η k is independent of the random variables η , . . . , η k , and thusof the sigma-algebra F k , we have the identities(5.16) X k − = E [ X k ∣ F k − ] = E [ ̃ X k ∣ F k ] . As a consequence of the definitions (5.14), (5.15) and the identities (5.16), we obtain E [( X k − ̃ X k ) ] = E [ X k + ̃ X k ] − E [ X k ̃ X k ] (5.17) = E [ X k ] − E [ X k X k − ] = E [( X k − X k − ) ] . We then decompose the random variable ⟨ φ ( )⟩ µ η Λ L according to the formula(5.18) ⟨ φ ( )⟩ µ η Λ L − E [⟨ φ ( )⟩ µ η Λ L ] = ( L + ) d − ∑ k = ( X k + − X k ) . Squaring the identity (5.18), integrating over the random field η , and using (5.17), we obtain(5.19) Var [⟨ φ ( )⟩ µ η Λ L ] = ( L + ) d − ∑ k = E [( X k + − X k ) ] = ( L + ) d ∑ k = E [( X k − ̃ X k ) ] . ANDOM-FIELD RANDOM SURFACES 21
We next prove the lower bound, for any integer k ∈ { , . . . , ( ⌊̃ cL ⌋ + ) d } ,(5.20) E [( X k − ̃ X k ) ] ≥ ⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩ λ c − cL d = ,λ c − c ( ln L ∨ ∣ x k ∣ ) d = ,λ cc − c ∨ ∣ x k ∣ d − d ≥ . Using that all the terms in the right side of (5.19) are non-negative, that the collection ofpoints x , . . . x d ⌊̃ cL ⌋ d ∨ is an enumeration of the points of the box Λ ̃ cL and the estimate (5.20)completes the proof of the lower bounds (1.8), (1.9) and (1.10).There remains to prove (5.20). Let us note that the two fields η and η k are equal on the setΛ L ∖ { x k } . We can thus apply Theorem 7 to obtain, on the event { η k ≠ ̃ η k } ,(5.21) ⟨ φ ( )⟩ µ η Λ L − ⟨ φ ( )⟩ µ ηk Λ L η k − ̃ η k ≥ ⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩ λc − cL d = ,λc − c ln ( L ∨ ∣ x k ∣ ) d = ,λc − c ( ∨ ∣ x k ∣ d − ) d ≥ . Using the formula (5.14) and integrating the inequality (5.21) over the field variables η k + , . . . , η L d ,we obtain, on the event { η k ≠ ̃ η k } ,(5.22) X k − ̃ X k η k − ̃ η k = ∫ ⟨ φ ( )⟩ µ η Λ L − ⟨ φ ( )⟩ µ ηk Λ L η k − ̃ η k ∏ j > k ν j ( dη j ) ≥ ⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩ λc − cL d = ,λc − c ln ( L ∨ ∣ x k ∣ ) d = ,λc − c ∨ ∣ x k ∣ d − d ≥ . The lower bound (5.22) shows, for any realization of the random fields η and ̃ η , ( X k − ̃ X k ) ≥ ⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩ λ c − cL ( η k − ̃ η k ) d = ,λ c − c ( η k − ̃ η k ) ( ln L ∨ ∣ x k ∣ ) d = ,λ c − c ( η k − ̃ η k ) ∨ ∣ x k ∣ d − d ≥ . Taking the expectation with respect to the external field, using that the two random variables η k and ̃ η k are independent and that their variance is equal to 1 completes the proof of (5.20). (cid:3) Integer-valued random-field Gaussian free field
In this section, we study the integer-valued random-field Gaussian free field. We prove thatits gradient delocalizes in one and two dimensions, and localizes in dimension 3 and higher, andthat its height delocalizes in one and two dimensions. Quantitative estimates are obtained ineach case.
Gradient fluctuations.
In this section, we prove the delocalization of the gradient of theinteger-valued Gaussian free field in the presence of a random field (Theorem 3). We establishthe following quenched result which implies Theorem 3 upon applying the Jensen inequalityand using the bounds in item (iv) of Proposition 4.1.
Theorem 8.
Let d ≥ , Λ ⊆ Z d be a finite subset of Z d , λ > be the strength of the randomfield, β > be an inverse temperature, η ∶ Λ → R be an external field and let u Λ ,η ∶ Λ + → R be the solution of (4.1) . Suppose φ is sampled from the probability distribution µ IV ,β,η Λ . Thenthere exists a constant C depending on the dimension d such that (6.1) ⟨ exp ( β ∥ ∇ φ − λ ∇ u Λ ,η ∥ L ( Λ + ) )⟩ µ IV ,β,η Λ ≤ exp ( C ( + β ) ∣ Λ + ∣) . Proof.
We first treat the specific case η = C depending on d such that(6.2) ⟨ exp ( β ∥ ∇ φ ∥ L ( Λ + ) )⟩ µ IV ,β, ≤ exp ( C ( + β ) ∣ Λ + ∣) . To prove the estimate (6.2), we first note that, for each φ ∶ Λ + → Z normalized to be 0 on theboundary ∂ Λ, the cardinality of the set { ψ ∶ Λ + → Z ∶ ψ ≡ ∂ Λ and ⌊ ψ / ⌋ = φ } is smallerthan 2 ∣ Λ ∣ . From this observation, one deduces the inequality ∑ ψ ∶ Λ + → Z exp ( − β ∥ ∇ ⌊ ψ ⌋∥ L ( Λ + ) ) = ∑ φ ∶ Λ + → Z ∑ ψ ∶ ⌊ ψ ⌋ = φ exp ( − β ∥ ∇ φ ∥ L ( Λ + ) ) (6.3) ≤ ∣ Λ ∣ ∑ φ ∶ Λ + → Z exp ( − β ∥ ∇ φ ∥ L ( Λ + ) ) . Using that the difference between a real number and its floor is smaller than 1, and using theinequality ( a + b ) ≤ a + b , we obtain ∥ ∇ ⌊ ψ ⌋∥ L ( Λ + ) = ∥ ∇ ψ + ( ∇ ⌊ ψ ⌋ − ∇ ψ )∥ L ( Λ + ) ≤ ∥ ∇ ψ ∥ L ( Λ + ) + C ∣ Λ + ∣ . Combining the inequalities (6.3) and (6.4) yields(6.4) ∑ φ ∶ Λ + → Z exp ( − β ∥ ∇ φ ∥ L ( Λ + ) ) ≤ ∣ Λ ∣ × e Cβ ∣ Λ + ∣ ∑ φ ∶ Λ + → Z exp ( − β ∥ ∇ φ ∥ L ( Λ + ) ) . We then write ⟨ exp ( β ∥ ∇ φ ∥ L ( Λ + ) )⟩ µ IV ,β,ηL = Z IV ,β, ∑ φ ∶ Z d → Z exp ( β ∥ ∇ φ ∥ L ( Λ + L ) ) × exp ( − β ∥ ∇ φ ∥ L ( Λ + L ) ) (6.5) = Z IV ,β, ∑ φ ∶ Z d → Z exp ( − β ∥ ∇ ψ ∥ L ( Λ + L ) ) . Using the explicit formula for the partition function Z IV ,β, ∶ = ∑ φ ∶ Λ + → Z exp ( − β ∥ ∇ φ ∥ L ( Λ + ) ) andcombining the inequality (6.4) with the identity (6.5) and the inequality ∣ Λ ∣ ≤ ∣ Λ + ∣ completesthe proof of (6.2).We now prove the inequality (6.1) for a general external field η ∶ Λ → R . Applying theestimate (6.2) at inverse temperature β , we see that it is sufficient to prove that there existsa constant C > d such that(6.6) ⟨ exp ( β ∥ ∇ φ − λ ∇ u Λ ,η ∥ L ( Λ + ) )⟩ µ IV ,β,η Λ ≤ exp ( Cβ ∣ Λ + ∣) ⟨ exp ( β ∥ ∇ φ ∥ L ( Λ + ) )⟩ µ IV , β , . ANDOM-FIELD RANDOM SURFACES 23
The rest of the argument is thus devoted to the proof of the inequality (6.6). Using the identity − ∆ u Λ ,η = η and performing a discrete integration by parts, one obtains, for any integer-valuedsurface φ ∶ Λ + → Z normalized to be 0 on ∂ Λ,12 ∥ ∇ φ ∥ L ( Λ + ) − λ ∑ x ∈ Λ φ ( x ) η ( x ) = ∥ ∇ φ ∥ L ( Λ + ) − λ ∑ e ∈ E ( Λ + ) ∇ φ ( e ) ∇ u Λ ,η ( e ) (6.7) = ∥ ∇ φ − λ ∇ u Λ ,η ∥ L ( Λ + ) − ∥ λ ∇ u Λ ,η ∥ L ( Λ + ) . Using the previous computation, we see that ⟨ exp ( β ∥ ∇ φ − λ ∇ u Λ ,η ∥ L ( Λ + ) )⟩ µ IV ,β,η Λ (6.8) = Z IV ,β,η Λ ∑ φ ∶ Λ + → Z exp ( β ∥ ∇ φ − λ ∇ u Λ ,η ∥ L ( Λ + ) ) exp ( − β ∥ ∇ φ − λ ∇ u Λ ,η ∥ L ( Λ + ) + β ∥ λ ∇ u Λ ,η ∥ L ( Λ + ) ) = Z IV ,β,η Λ × exp ( − β ∥ ∇ u Λ ,η ∥ L ( Λ L ) ) ∑ φ ∶ Λ + → Z exp ( − β ∥ ∇ φ − λ ∇ u Λ ,η ∥ L ( Λ + ) ) . Additionally, from the definition of the partition function Z IV ,β,η Λ L and the computation (6.7),we obtain the identity Z IV ,β,η Λ × exp ( − β ∥ ∇ u η ∥ L ( Λ ) ) = ∑ φ ∶ Λ → Z exp ( − β ∥ ∇ φ ∥ L ( Λ + ) + βλ ∑ x ∈ Λ φ ( x ) η ( x ) − β ∥ λ ∇ u Λ ,η ∥ L ( Λ ) ) (6.9) = ∑ φ ∶ Λ + → Z exp ( − β ∥ ∇ φ − λ ∇ u Λ ,η ∥ L ( Λ + ) ) . Let us then introduce the mapping w Λ ,η ∶ = λu Λ ,η − ⌊ λu Λ ,η ⌋ . Since the function ⌊ λu Λ ,η ⌋ isinteger-valued, we may perform the discrete change of variable φ ↦ φ − ⌊ λu Λ ,η ⌋ , and write(6.10) ∑ φ ∶ Λ + → Z exp ( − β ∥ ∇ φ − λ ∇ u Λ ,η ∥ L ( Λ + ) ) = ∑ φ ∶ Λ + → Z exp ( − β ∥ ∇ φ − ∇ w Λ ,η ∥ L ( Λ + ) ) . By the definition of the mapping w Λ ,η , it is bounded by 1. Applying the inequality ( a + b ) ≥ a − b , we obtain(6.11) ∥ ∇ φ − ∇ w Λ ,η ∥ L ( Λ + ) ≥ ∥ ∇ φ ∥ L ( Λ + ) − C ∣ Λ + ∣ . A combination of (6.10) and (6.11) implies(6.12) ∑ φ ∶ Λ + → Z exp ( − β ∥ ∇ φ − λ ∇ u Λ ,η ∥ L ( Λ + ) ) ≤ exp ( Cβ ∣ Λ + ∣) ∑ φ ∶ Λ + → Z exp ( − β ∥ ∇ φ ∥ L ( Λ + ) ) . A similar computation, but using this time the inequality ∥ ∇ φ − ∇ w Λ ,η ∥ L ( Λ + ) ≤ ∥ ∇ φ ∥ L ( Λ + ) + C ∣ Λ + ∣ , yields the lower bound ∑ φ ∶ Λ + → Z exp ( − β ∥ ∇ φ − λ ∇ u Λ ,η ∥ L ( Λ + ) ) = ∑ φ ∶ Λ + → Z exp ( − β ∥ ∇ φ − ∇ w Λ ,η ∥ L ( Λ + ) ) (6.13) ≥ exp ( − Cβ ∣ Λ + ∣) ∑ φ ∶ Λ + → Z exp ( − β ∥ ∇ φ ∥ L ( Λ + ) ) . A combination of the identities (6.8), (6.9) with the inequalities (6.12) and (6.13) implies theupper bound ⟨ exp ( β ∥ ∇ φ − λ ∇ u Λ ,η ∥ L ( Λ + ) )⟩ µ IV ,β,η Λ ≤ exp ( Cβ ∣ Λ + ∣) ∑ φ ∶ Λ + → Z exp ( − β ∥ ∇ φ ∥ L ( Λ + ) ) ∑ φ ∶ Λ + → Z exp ( − β ∥ ∇ φ ∥ L ( Λ + ) ) = exp ( Cβ ∣ Λ + ∣) ⟨ exp ( β ∥ ∇ φ ∥ L ( Λ + ) )⟩ µ IV , β , . The proof of the inequality (6.6) is complete. (cid:3)
Height fluctuations in one and two dimensions.
The purpose of this section is toprove that the integer-valued Gaussian free field delocalizes in the presence of a random fieldin dimensions 1 and 2. The proof is decomposed into two statements: we first establish inLemma 6.1 a quenched inequality satisfied by the thermal expectation of the discrete Gaussianfree field, and then use this inequality to prove the lower bound of Theorem 4.
Lemma 6.1.
Let d ≥ , Λ ⊆ Z d be a finite subset of Z d , λ > be the strength of the randomfield, β > an inverse temperature, η ∶ Λ → R be an external field. Let φ be distributed accordingto the random-field integer-valued Gaussian free field µ IV ,β,η Λ . Then for any map w ∶ Λ + → Z normalized to be on the boundary ∂ Λ , (6.14) 2 ∑ e ∈ E ( Λ + ) ⟨ ∇ φ ( e )⟩ µ IV ,β,η Λ ∇ w ( e ) − λ ∑ x ∈ Λ η ( x ) w ( x ) + ∥ ∇ w ∥ L ( Λ + ) ≥ . Proof.
We only prove the inequality (6.14) in the case η =
0. The general case can be obtainedby a notational modification of the argument. By performing the discrete change of variables φ → φ + w , we have the identity(6.15) ∑ φ ∶ Λ → Z exp ( − β ∥ ∇ φ ∥ L ( Λ + ) ) = ∑ φ ∶ Λ → Z exp ( − β ∥ ∇ φ + ∇ w ∥ L ( Λ + ) ) . Subtracting the right and left hand sides of the identity (6.15), and expanding the square, weobtain the identity ∑ φ ∶ Λ → Z exp ( − β ∥ ∇ φ ∥ L ( Λ + ) ) ⎛⎝ exp ⎛⎝ − β ∑ e ∈ E ( Λ + ) ∇ φ ( e ) ∇ w ( e ) − β ∥ ∇ w ∥ L ( Λ + ) ⎞⎠ − ⎞⎠ = . Using the identity e y − ≥ y , we obtain the estimate(6.16) ∑ φ ∶ Λ → Z exp ( − β ∥ ∇ φ ∥ L ( Λ + ) ) ⎛⎝ − ∑ e ∈ E ( Λ + ) ∇ φ ( e ) ∇ w ( e ) − ∥ ∇ w ∥ L ( Λ + ) ⎞⎠ ≤ . Dividing both sides of the inequality (6.16) by the partition function Z IV ,β, completes theproof of the inequality (6.14). (cid:3) We are now ready to give the proof of Theorem 4.
Proof of Theorem 4.
Fix a dimension d ∈ { , } and an integer L ≥
1. We first prove the upperbounds of (1.18) and (1.19). The proof relies on the result of Theorem 8 which, combined withthe Jensen inequality, yields the L -estimate ∥ ∇ φ − λ ∇ u Λ L ,η ∥ L ( Λ + L ,µ IV ,β,η Λ L ) ≤ C ( + β ) . ANDOM-FIELD RANDOM SURFACES 25
Since φ and the mapping u Λ L ,η are normalized to be 0 on the boundary ∂ Λ L , one can applythe Poincar´e inequality and obtain(6.17) ∥ φ − λu Λ L ,η ∥ L ( Λ + L ,µ IV ,β,η Λ L ) ≤ CL ∥ ∇ φ − λ ∇ u Λ L ,η ∥ L ( Λ + L ,µ IV ,β,η Λ L ) ≤ C ( + β ) L. Combining the inequality (6.17) with the bounds in item (iv) of Proposition 4.1 completes theproof of the upper bound of Theorem 4.We now prove the lower bound of Theorem 4. Let us first define the averaged field t Λ L ∶ = ∣ Λ L ∣ − ∑ y ∈ Λ L η ( y ) and the orthogonal field η ⊥ by the formula, for any vertex x ∈ Λ L , η ⊥ ( x ) ∶ = η ( x ) − t Λ L . Under the assumption that the random field ( η ( x )) x ∈ Λ L is a collection of independentand normalized Gaussian random variables, we see that t Λ L is Gaussian, that its variance isequal to ∣ Λ L ∣ − , and that the random variable t Λ L and the field η ⊥ are independent.For each integer L ≥
1, we define the map P L ∶ Z d → Z by the formula, for each x ∈ Z d ,(6.18) P L ( x ) ∶ = ⌊ L − d ⌋ inf z ∈ Z d ∖ Λ L / ∣ x − z ∣ . For each vertex y ∈ Λ L / , we denote by P L,y ∶ = P L ( y + ⋅ ) .Let us fix a point y ∈ Z d and record some of the properties of the functions P L,y . First the P L,y is non-negative, integer-valued and its support is included in the box y + Λ L / . Moreoverthere exists a universal constant c > P L,y satisfies the estimates(6.19) ∑ x ∈ Λ L P L,y ( x ) ≥ cL + d and ∣ ∇ P L,y ∣ ≤ L − d . Consequently, there exists a universal constant C P such that, for each integer L ≥ ∥ ∇ P L,y ∥ L ( Λ + L ) − C P L − d ∑ x ∈ Λ L P L,y ( x ) ≤ − L . We finally note that the discrete Laplacian of the map P L,y satisfies the pointwise upper bound ∣ ∆ P L,y ∣ ≤ CL − d . Additionally, if we denote by C ∶ = supp ∆ P L, the support of the map ∆ P L, ,then we have supp P L,y = y + C . The set C is explicit: d = ∶ C ∶ = { − ⌊ L ⌋ , , ⌊ L ⌋} ,d = ∶ C ∶ = ∂ Λ L / ∪ {( x, x ) ∶ x ∈ { − ⌊ L ⌋ , . . . , ⌊ L ⌋}} ∪ {( x, − x ) ∶ x ∈ { − ⌊ L ⌋ , . . . , ⌊ L ⌋}} . In dimension 2, this set corresponds to the boundary of the square Λ L / and its two diagonals.We finally note that there exists a constant c > ∣ C ∣ ≥ cL d − .For each realization of the orthogonal field η ⊥ , we denote by S ∶ = S ( η ⊥ ) ⊆ Λ L / the randomset S ∶ = ⎧⎪⎪⎨⎪⎪⎩ y ∈ Λ L / ∶ ∑ x ∈ Λ L P L,y ( x ) η ⊥ ( x ) ≥ ⎫⎪⎪⎬⎪⎪⎭ . We note that the set S depends only on the realization of the orthogonal field η ⊥ , and is thusindependent of the averaged field t Λ . Additionally, it satisfies the almost sure inequality(6.20) Card S ( η ⊥ ) + Card S ( − η ⊥ ) ≥ ∣ Λ L / ∣ . Since the law of the orthogonal field η ⊥ is invariant under the involution η ⊥ ↦ − η ⊥ , the inequal-ity (6.20) implies the lower bound P ⎛⎝ Card S ≥ ∣ Λ L / ∣ ⎞⎠ ≥ . We claim that Theorem 4 is a consequence of the following inclusion of events: there exists auniversal constant c > { t Λ ≥ C P λL d } ∩ ⎧⎪⎪⎨⎪⎪⎩ Card S ≥ ∣ Λ L / ∣ ⎫⎪⎪⎬⎪⎪⎭ ⊆ ⎧⎪⎪⎨⎪⎪⎩ ∣ Λ L ∣ ∑ v ∈ Λ L ∣⟨ φ ( v )⟩ µ IV ,β,η Λ L ∣ ≥ cL − d ⎫⎪⎪⎬⎪⎪⎭ . To see that (6.21) implies Theorem 4, let us note that the independence of the random variables t Λ L and Card S and the fact that the averaged field t Λ L is a Gaussian random variable of variance ∣ Λ L ∣ − = L − d imply the inequalitylim inf L →∞ P ⎡⎢⎢⎢⎢⎣ ∣ Λ L ∣ ∑ v ∈ Λ L ∣⟨ φ ( v )⟩ µ η Λ L ∣ ≥ cL − d ⎤⎥⎥⎥⎥⎦ ≥ lim inf L →∞ P [ t Λ L ≤ − C P λL d ] P ⎡⎢⎢⎢⎣ Card S ≥ ∣ Λ L / ∣ ⎤⎥⎥⎥⎦ (6.22) ≥ √ π ∫ − CPλ −∞ e − t dt × > ce − cλ . The lower bound (6.22) and the Jensen inequality imply Theorem 4.We now prove the inclusion (6.21). To this end, let us fix a realization of the random field η such that t Λ L ≤ − C P λ − L − d and Card S ≥ ∣ Λ L / ∣ /
2, and fix a point y ∈ S . Using Lemma 6.1with the map w = P L,y , we obtain the inequality0 ≤ ∑ e ∈ E ( Λ + L ) ⟨ ∇ φ ( e ) ∇ P L,y ( e )⟩ µ IV ,β,η Λ L + ∥ ∇ P L,y ∥ L ( Λ + L ) − λ ∑ v ∈ Λ L η ( x ) P L,y ( x ) . Using the definition of the averaged and orthogonal fields t Λ L and η ⊥ , the previous inequalitycan be equivalently rewritten0 ≤ ∑ e ∈ E ( Λ + L ) ⟨ ∇ φ ( e ) ∇ P L,y ( e )⟩ µ IV ,β,η Λ L + ∥ ∇ P L,y ∥ L ( Λ + L ) − λt Λ L ∑ x ∈ Λ L P L,y ( x ) − λ ∑ x ∈ Λ L η ( x ) ⊥ P L,y ( x ) . Using the assumptions y ∈ S , t Λ L ≥ C P λ − L − d / , and the properties of the map P L,y , we obtainthe inequality2 L ≤ − ∥ ∇ P L,y ∥ L ( Λ + L ) + C P L d ∑ x ∈ Λ L P L,y ( x ) ≤ ∑ e ∈ E ( Λ + L ) ⟨ ∇ φ ( e ) ∇ P L,y ( e )⟩ µ IV ,β,η Λ L . Performing a discrete integration by parts on the term in the right-hand side, we obtain thelower bound − ∑ x ∈ Λ L ⟨ φ ( x ) ∆ P L,y ( x )⟩ µ IV ,β,η Λ L ≥ L . Using that the Laplacian of the map P L,y is bounded by CL − d and supported in the set y + C ,we obtain the lower bound ∑ x ∈ y + C ∣⟨ φ ( x )⟩ µ IV ,β,η Λ L ∣ ≥ L + d . We then use that the cardinality of the set C is larger than cL d − and obtain the estimate1 ∣ C ∣ ∑ x ∈ y + C ∣⟨ φ ( x )⟩ µ IV ,β,η Λ L ∣ ≥ cL − d . Summing over all the points y ∈ S and dividing by the cardinality of S , we obtain the inequality1 ∣ S ∣ ∑ y ∈S ∣ C ∣ ∑ x ∈ y + C ∣⟨ φ ( x )⟩ µ IV ,β,η Λ L ∣ ≥ cL − d . ANDOM-FIELD RANDOM SURFACES 27
Using the assumption ∣ S ∣ ≥ ∣ Λ L / ∣ /
2, that the cardinality of the set C is larger than cL d − , weobtain the estimate(6.23) 1 L d − ∑ y ∈S ∑ x ∈C ∣⟨ φ ( y + x )⟩ µ IV ,β,η Λ L ∣ ≥ cL − d . The estimate (6.23) can be equivalently rewritten1 L d ∑ z ∈ Λ L N C , S ( z ) L d − ∣⟨ φ ( z )⟩ µ IV ,β,η Λ L ∣ ≥ cL − d , where the number N C , S ( z ) is defined by the formula(6.24) N C , S ( z ) ∶ = Card {( x, y ) ∈ C × S ∶ x + y = z } = Card { y ∈ S ∶ z ∈ y + C } . We then fix a point z ∈ Λ L and estimate the number N C , S ( z ) . To this end, we note that theset C is symmetric, and thus, for any pair of points y, z ∈ Λ L , z ∈ y + C ⇐⇒ y ∈ z + C . This equivalence implies the identity { y ∈ S ∶ z ∈ y + C } ⊆ z + C , which implies the upper bound N C ( z ) ≤ CL d − . Combining this result with the inequality (6.24)shows 1 L d ∑ z ∈ Λ L ∣⟨ φ ( z )⟩ µ IV ,β,η Λ L ∣ ≥ cL − d . Using that L d ≤ ( L + ) d = ∣ Λ L ∣ together with the previous inequality completes the proof ofthe inclusion of events (6.21). (cid:3) Height fluctuations in dimensions d ≥ . The objective of this section is to proveTheorem 5. The argument relies on a Peierls-type argument, and we will make use of the twofollowing propositions. The first one is a result of Fisher–Fr¨ohlich–Spencer [27].
Proposition 6.2 (Fisher–Fr¨ohlich–Spencer [27]) . Let d ≥ and ( η ( x )) x ∈ Z d be a collection ofindependent standard Gaussian random variables. There exist two constants c, λ ′ > such thatfor any λ ∈ ( , λ ′ ] and any v ∈ Z d , the event (6.25) E λ,v ∶ = { η ∶ Z d → R ∶ ∀ Λ ⊆ Z d such that Λ is connected, bounded and v ∈ Λ , λ ∣ ∑ x ∈ Λ η ( x )∣ ≤ ∣ ∂ Λ ∣} satisfies P ( E λ,v ) ≥ − e − cλ . We mention that the proof of Fisher–Fr¨ohlich–Spencer [27] is written in the case d = d ≥ Z d containing a fixed vertex v and with fixed volume boundary. The resultcan found in Bollob´as [7] (see also [42, 6]) Proposition 6.3 (Corollary 1.2 of [42]) . Let d ≥ . There exists a constant C d depending onthe dimension d such that, for each integer N ∈ N and each v ∈ Z d , the set A N,v ∶ = { Λ ⊆ Z d ∶ v ∈ Λ , Λ is connected, bounded and ∣ ∂ Λ ∣ = N } satisfies ∣ A N,v ∣ ≤ e C d N . We are now ready to give the proof of Theorem 5.
Proof of Theorem 5.
Fix d ≥ L ≥ v ∈ Λ L . Let λ ′ and C d be theconstants appearing in the statements of Propositions 6.2 and 6.3, and, for λ ∈ ( , λ ′ ) , let E λ,v be the event (6.25). Let us set λ = λ ′ / β = C d . We fix λ ∈ ( , λ ] and β ∈ [ β , ∞ ) . ByProposition 6.3, it is sufficient, in order to obtain Theorem 5, to prove the inclusion of events(6.26) { η ∶ ⟨ { φ ( v ) = } ⟩ µ IV ,β,η Λ L ≥ − Ce − β } ⊆ E λ,v . We now prove (6.26). To this end, let us fix a realization of the random field η such that η ∈ E λ,v , and let φ ∶ Λ L → Z be a random surface distributed according to the random-fieldinteger-valued Gaussian free field µ IV ,β,η Λ L . We denote by D + ( φ ) the connected component of v in the set { x ∈ Λ L ∶ φ ( x ) ≥ } , and set D + ( φ ) = ∅ if φ ( v ) ≤
0. Symmetrically, we let D − ( φ ) bethe connected component of v in the set { x ∈ Λ L ∶ φ ( x ) ≤ − } , and set D − ( φ ) = ∅ if φ ( v ) ≥ φ ( v ) = ⇐⇒ D + ( φ ) = ∅ and D − ( φ ) = ∅ . We next claim that there exists a constant C ≥ η ∈ E λ,v ,(6.28) ⟨ { D + ( φ ) ≠ ∅} ⟩ µ IV ,β,η Λ L + ⟨ { D − ( φ ) ≠ ∅} ⟩ µ IV ,β,η Λ L ≤ Ce − β . A combination of (6.27) and (6.28) yields ⟨ { φ ( v ) = } ⟩ µ IV ,β,η Λ L = − ⟨ { φ ( ) ≠ } ⟩ µ IV ,β,η Λ L ≥ − ⟨ { D + ( φ ) ≠ ∅} ⟩ µ IV ,β,η Λ L − ⟨ { D + ( φ ) ≠ ∅} ⟩ µ IV ,β,η Λ L ≥ − Ce − β , which yields the inclusion (6.26). The rest of the proof is devoted to the demonstration of (6.28).By a symmetry argument, it is sufficient to prove(6.29) ⟨ { D + ( φ ) ≠ ∅} ⟩ µ IV ,β,η Λ L ≤ Ce − β . To prove the estimate (6.29), we first decompose the event { D + ( φ ) ≠ ∅ } according to theidentity(6.30) ⟨ { D + ( φ ) ≠ ∅} ⟩ µ IV ,β,η Λ L = ∑ D ⊆ Λ L ⟨ { D + ( φ ) = D } ⟩ µ IV ,β,η Λ L , where the sum in the right-hand side is computed over all the bounded connected subsets D containing the vertex 0.Let us fix a bounded connected subset D ⊆ Λ L such that 0 belongs to D . We have theidentity(6.31) ⟨ { D + ( φ ) = D } ⟩ µ IV ,β,η Λ L = ∑ φ ∶ Λ L → Z D + ( φ ) = D exp ( − β ∥ ∇ φ ∥ L ( Λ + L ) + βλ ∑ x ∈ Λ L η ( x ) φ ( x )) ∑ φ ∶ Λ L → Z exp ( − β ∥ ∇ φ ∥ L ( Λ + L ) + βλ ∑ x ∈ Λ L η ( x ) φ ( x )) . For each integer-valued φ ∶ Λ L → Z such that D + ( φ ) = D , we define ̃ φ according to the formula ̃ φ ∶ = φ − D . Using that, by the identity D + ( φ ) = D , the map φ satisfies φ ≥ D and φ ≤ ∂D , we have the inequality, for any edge e ∈ E ( Λ + L ) ,(6.32) ( ∇ ̃ φ ( e )) = ( ∇ φ ( e )) − ( ∣ ∇ φ ( e )∣ − ) ∣ ∇ D ( e )∣ ≤ ( ∇ φ ( e )) − ∣ ∇ D ( e )∣ . Summing the inequality (6.32) over the edges e ∈ E ( Λ + L ) and using that the cardinality of thesupport of the mapping ∇ D is a least ∣ ∂D ∣ (since an edge belongs to the support this map if ANDOM-FIELD RANDOM SURFACES 29 and only if it has exactly one endpoint in D and one endpoint in ∂D ), we obtain(6.33) ∥ ∇ ̃ φ ∥ L ( Λ + L ) ≤ ∥ ∇ φ ∥ L ( Λ + L ) − ∣ ∂D ∣ . Combining the inequality (6.33) with the assumption η ∈ E λ,v , we obtain12 ∥ ∇ ̃ φ ∥ L ( Λ + L ) − λ ∑ x ∈ Λ L η ( x )̃ φ ( x ) = ∥ ∇ ̃ φ ∥ L ( Λ + L ) − λ ∑ x ∈ Λ L η ( x ) φ ( x ) + λ ∑ x ∈ D η ( x ) ≤ ∥ ∇ φ ∥ L ( Λ + L ) − λ ∑ x ∈ Λ L η ( x ) φ ( x ) − ∣ ∂D ∣ + λ ∑ x ∈ D η ( x ) ≤ ∥ ∇ φ ∥ L ( Λ + L ) − λ ∑ x ∈ Λ L η ( x ) φ ( x ) − ∣ ∂D ∣ . Using the previous computation, we may write ∑ φ ∶ Λ L → Z D + ( φ ) = D exp ⎛⎝ − β ∥ ∇ φ ∥ L ( Λ + L ) + βλ ∑ x ∈ Λ L η ( x ) φ ( x )⎞⎠ (6.34) ≤ exp ( − β ∣ ∂D ∣ ) ∑ φ ∶ Λ L → Z D + ( φ ) = D exp ⎛⎝ − β ∥ ∇ ̃ φ ∥ L ( Λ + L ) + βλ ∑ x ∈ Λ L η ( x )̃ φ ( x )⎞⎠ ≤ exp ( − β ∣ ∂D ∣ ) ∑ φ ∶ Λ L → Z exp ⎛⎝ − β ∥ ∇ φ ∥ L ( Λ + L ) + βλ ∑ x ∈ Λ L η ( x ) φ ( x )⎞⎠ . A combination of the identity (6.31) and the inequality (6.34) implies(6.35) ⟨ { D + ( φ ) = D } ⟩ µ IV ,β,η Λ L ≤ exp ( − β ∣ ∂D ∣ ) . Putting the estimate (6.35) back into (6.30), using Proposition 6.3, the assumption β ≥ β andthe identity β ∶ = C d (which imply β / − C d ≥ β / ⟨ { D + ( φ ) ≠ ∅} ⟩ µ IV ,β,η Λ L ≤ ∑ D ⊆ Λ L e − β ∣ ∂D ∣ ≤ L d ∑ N = ∣ A N,v ∣ e − βN ≤ L d ∑ N = e C d N e − βN ≤ Ce − β . The proof of (6.29), and thus of Theorem 5, is complete. (cid:3) Discussion and open questions
In this section we highlight several research directions.7.1.
The integer-valued random-field Gaussian free field and the random-phase sine-Gordon model.
Theorem 3 determines that the gradient of the integer-valued random-fieldGaussian free field delocalizes in dimensions d = , d ≥
3, at alltemperatures (including zero temperature) and all positive disorder strengths λ . Our resultsfor the height fluctuations are less complete: Theorem 4 proves delocalization in dimensions d = ,
2, again at all temperatures and positive disorder strengths, while Theorem 5 establisheslocalization in dimensions d ≥ d = ,
4; see Theorem 2). The height fluctuations indimensions d ≥ d ≥
3, temperature and disorder strength would constitutea roughening transition from the localized behavior proved in Theorem 5 and would thus be ofsignificant interest. Unlike the non-disordered case, where roughening transitions are familiar only in two dimensions [28, 29, 51, 40], we conjecture that such a transition indeed takes placefor the three-dimensional integer-valued random-field Gaussian free field, with delocalizationoccurring at least in the low-temperature and strong disorder regime. We further conjecturethat no transition occurs in dimensions d ≥
5, where we expect the surface to remain localized atall temperatures and disorder strengths (as in the real-valued case; see Theorem 2). These con-jectures are supported by a connection of the model with the random-phase sine-Gordon modelon which we elaborate next. We do not conjecture regarding the intermediate four-dimensionalcase.The sine-Gordon model is a model of real-valued surfaces ψ whose Hamiltonian, on a domainΛ with given boundary conditions, takes the form(7.1) ∑ e ∈ E ( Λ + ) ( ∇ ψ ( e )) + z ∑ v ∈ Λ cos ( π ( ψ ( v ) − r ( v ))) , where the ( r ( v )) v ∈ Λ are given elements of the torus R / Z and z ≥ z → ∞ configurations ψ are restricted to satisfying ψ v ∈ Z + r ( v ) at every vertex andtheir effective Hamiltonian consists only of the first sum in (7.1). The case where the ( r ( v )) are (quenched) random is known as the random-phase sine-Gordon model .The random-phase sine-Gordon model has received much attention in the physics literature(see, e.g., [15, 49, 35, 41, 47]) and the following behavior was predicted for the heights ψ in the z → ∞ limit when the ( r ( v )) are uniform and independent: In two dimensions, on a box Λ L withzero boundary conditions, the heights are predicted to delocalize with log L variance at hightemperature (rough phase), but with log L variance at low temperature (super-rough phase);thus the fluctuations of the model are expected to decrease as the temperature rises! It isfurther predicted that the heights delocalize with logarithmic variance in three dimensions andare localized when d ≥ ( r ( v )) and any activity z ∈ [ , ∞ ] , the heights ψ on a two-dimensional box Λ L with zero boundaryconditions delocalize with variance at least log L .Let us now describe the connection between the random-field integer-valued Gaussian freefield and the random-phase sine-Gordon model. To this end, observe that the Hamiltonianof the random-field integer-valued Gaussian free field can be written, up to the addition of aconstant factor depending only on d, λ and the quenched disorder η , as12 ∑ e ∈ E ( Λ + ) ( ∇ φ ( e ) − λ ∇ u Λ ,η ( e )) where we recall that u Λ ,η , the ground state of the real-valued random-field Gaussian free field, isdefined in (4.1). Let φ be sampled from the integer-valued random-field Gaussian free field anddefine ψ ∶ = φ − λu Λ ,η . Observe that ψ ( v ) ∈ Z + r ( v ) at every vertex where r ( v ) ∶ = − λ proj u Λ ,η ( v ) and where proj denotes the canonical projection from R to R / Z . It follows that ψ is distributedas the random-phase sine-Gordon model in the z → ∞ limit with this choice of r ( v ) (notingthat r is a function of the quenched disorder η ).The distribution of the vector r in the above connection is not uniform and independent,though for large λ we expect it to be ‘somewhat close’ to such a distribution, at least when the ( η v ) are independent standard Gaussian random variables. It is thus suggested that for large λ the distribution of ψ is close to that of the random-phase sine-Gordon model with uniformand independent r . This supports the above conjectures regarding the height fluctuations ofthe integer-valued random-field Gaussian free field in the strong disorder regime.7.2. The random-field membrane model.
Another related model of real-valued randomsurfaces is the membrane model. In this setting, given a finite domain Λ ⊆ Z d , an external field ANDOM-FIELD RANDOM SURFACES 31 η ∶ Λ → R , the noised Hamiltonian of a finite-volume φ ∶ Z d → R normalized to 0 on ∂ Λ is givenby the formula H η, ∆Λ ( φ ) ∶ = ∥ ∆ φ ∥ L ( Λ ) − ∑ x ∈ Λ η ( x ) φ ( x ) . The random-field membrane model is the probability distribution(7.2) µ η, ∆Λ ( dφ ) ∶ = Z η, ∆Λ exp ( − H η, ∆Λ ( φ )) , where the partition function Z η, ∆Λ is the constant which makes the measure (7.2) a probabilitydistribution. Let us first consider the ground state of the model, that is, the interface v Λ ,η ∶ Λ L → R which minimizes the variational probleminf w ∶ Z d → R w ≡ Z d ∖ Λ ∥ ∆ φ ∥ L ( Λ ) − ∑ x ∈ Λ η ( x ) φ ( x ) . Note that it can can be equivalently defined as the solution of the biharmonic equation ⎧⎪⎪⎨⎪⎪⎩ ∆ v Λ ,η ( x ) = η in Λ ,v Λ ,η = Z d ∖ Λ . In that case, the mapping v Λ ,η is a linear functional of η , and is given by the explicit formula v Λ ,η ( x ) = ∑ y,z ∈ Λ L G Λ L ( x, y ) G Λ L ( y, z ) η ( z ) . Using the upper and lower bounds on the Green’s function stated in (4.1), we obtain that theground state satisfies the following estimates1 ≤ d ≤ ∶ cL − d ≤ E [ v Λ ,η ( ) ] ≤ CL − d ,d = ∶ c ln L ≤ E [ v Λ ,η ( ) ] ≤ C ln L,d ≥ ∶ c ≤ E [ v Λ ,η ( ) ] ≤ C. It is thus delocalized in dimensions d ≤ d ≥ . To study the probability distribution (7.2), let us observe that if φ ∶ Λ L → R is a randomsurface distributed according to (7.2), then ψ ∶ = φ − v Λ ,η is a membrane model with externalfield set to 0. Two consequences can be deduced from this observation: first, the expectationof the random variable ψ is equal to 0, and second the fluctuations of ψ at the center of thebox can be explicitly quantified and we have, for any realization of the random field η ,1 ≤ d ≤ ∶ cL − d ≤ ⟨ ψ ( ) ⟩ µ η, ∆Λ ≤ CL − d ,d = ∶ c ln L ≤ ⟨ ψ ( ) ⟩ µ η, ∆Λ ≤ C ln L,d ≥ ∶ c ≤ ⟨ ψ ( ) ⟩ µ η, ∆Λ ≤ C. A combination of the two previous sets of estimates shows that the random-field membranemodel satisfies the inequalities1 ≤ d ≤ ∶ cL − d ≤ E [⟨ φ ( ) ⟩ µ η, ∆Λ ] ≤ CL − d ,d = ∶ c ln L ≤ E [⟨ φ ( ) ⟩ µ η, ∆Λ ] ≤ C ln L,d ≥ ∶ c ≤ E [⟨ φ ( ) ⟩ µ η, ∆Λ ] ≤ C. The model is localized in dimensions d ≥ d ≤ Scaling limit of the random-field ∇ φ -model. Theorem 2 establishes quantitativeestimates on the height of the random-field ∇ φ -model. A first open question we mentionwould be to prove the existence of translation-covariant infinite-volume Gibbs measures for themodel in dimensions d ≥ gradient Gibbs measures isestablished in [19]), and to identify the scaling limit of these measures.A specific example of interest where the existence of the infinite-volume translation-covariantGibbs measure can be established and the scaling limit can be identified is the random-fieldGaussian free field when the external field is assumed to be i.i.d. with Gaussian distribution,since in that case explicit computations are available. To be more specific, for any boundeddomain Λ ⊆ Z d , and any realization of the external field η ∶ Λ → R , the Gaussian free field withexternal field η is the probability distribution(7.3) µ η, GFFΛ ( dφ ) ∶ = Z η, GFFΛ exp ( − ∥ ∇ φ ∥ L ( Λ + ) + ∑ x ∈ Λ η ( x ) φ ( x )) , where the partition function Z η, GFFΛ is chosen such that (7.3) is a probability measure. Let usrecall the definition of the mapping u Λ ,η stated in (4.1), and observe that, for any realizationof the random field η , if φ is distributed according to the Gaussian free field (7.3), then therandom surface φ − u Λ ,η is a Gaussian free field with external field equal to 0. In the specificsetting when the external field η consists of independent standard Gaussian random variables,the law of the mapping u η is Gaussian, and its convariance matrix is given by(7.4) ∀ x, y ∈ Λ , E [ u Λ ,η ( x ) u Λ ,η ( y )] = ∑ z ∈ Λ G Λ ( x, z ) G Λ ( z, y ) . One can additionally observe that the mapping v ( x, y ) ∶ = ∑ z ∈ Λ G Λ ( x, z ) G Λ ( z, y ) solves thediscrete biharmonic equation(7.5) ⎧⎪⎪⎨⎪⎪⎩ ∆ v ( ⋅ , y ) = δ y in Λ ,v ( ⋅ , y ) = Z d ∖ Λ . A consequence of the identities (7.4) and (7.5) is that the law of the ground state u Λ ,η is theone of the membrane model (i.e., the measure (7.2) with external field set to 0). Using that φ − u Λ ,η is distributed according to a Gaussian free field together with the article of Cipriani,Dan and Hazra [18] which identifies the scaling limit of the membrane model, one can showthat the scaling limit of the random-field Gaussian free field is a continuous membrane model inthe sense that, for any smooth functions f compactly supported in the continuous box [ − , ] d , L − d − ∑ x ∈ Λ L φ ( x ) f ( xL ) ( dist ) Ð→ N →∞ N ( , ∫ [ − , ] d f ( y ) w f ( y ) dy ) , where the convergence holds in distribution over the annealed measure P µ η, GFFΛ , and the map-ping w f is the solution of the continuous biharmonic equation ⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩ ∆ w f = f in Λ ,w f = ∂ Λ , n ⋅ ∇ w f = ∂ Λ . We may expect that a conclusion of similar nature should hold for the uniformly convex random-field random surface model (1.3).
ANDOM-FIELD RANDOM SURFACES 33
Random-field Gaussian free field with -dependent external field. Theorem 2states that, if the components of the random field η are independent, then the random-field ∇ φ -model is localized in dimensions d ≤ d ≥
5. One can thus raisethe question whether the independence assumption on the random field can be relaxed withoutchanging the critical dimension above which the random surface is localized and below whichit is delocalized. In this section, we present a model of a random-field random surface in whichthe external random field is 1-dependent, which localizes in dimensions d ≥ L ≥ ζ ∶ Λ L → R a discrete Gaussian free field in the box Λ L with Dirichlet boundary condition. We then define the external field η by the formula, for anyvertex x ∈ Z d ,(7.6) η ( x ) ∶ = − ∆ ζ ( x ) . The law of the external field η is Gaussian, let us verify that it is 1-dependent. To this end,we fix two points x, y ∈ Z d such that ∣ x − y ∣ ≥
1. An explicit computation shows, for any point z ∈ Λ L ,(7.7) E [ η ( x ) ζ ( z )] = E [ − ∆ ζ ( x ) ζ ( z )] = − ∆ G Λ L ( x, z ) = { x = z } . Using the identity (7.7) and an explicit computation, we deduce E [ η ( x ) η ( y )] = d { x = y } − ∑ z ∼ y { x = z } . Under the assumption ∣ x − y ∣ ≥
1, all the terms in the right-hand side are equal to 0 and thus E [ η ( x ) η ( y )] =
0. Since the field η is Gaussian, we deduce that the random variables η ( x ) and η ( y ) are independent.Let us now consider the random-field Gaussian free field with the external field η givenby (7.6). Since the random surface φ − ζ is a Gaussian free field with external field set to 0, wehave, for any deterministic realization of the external field,(7.8) ⟨( φ ( ) − ζ ( )) ⟩ µ η, GFFΛ ≤ ⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩ CL if d = ,C ln L if d = ,C if d ≥ . From the inequality (7.8) and the fact that ζ is a Gaussian free field with Dirichlet boundarycondition in the box Λ L , we obtain E [⟨ φ ( ) ⟩ µ η, GFFΛ ] ≤ ⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩ CL if d = ,C ln L if d = ,C if d ≥ . The random surface is thus localized in dimensions 3 and higher.7.5.
Relation between between the ground state of the ∇ φ model and its expectation. If we let φ be a random-field Gaussian free field with quenched disorder η , then we have theidentity, for any sidelength L ≥ x ∈ Λ L , ⟨ φ ( x )⟩ µ η Λ L = u Λ ,η ( x ) , that is, the thermal expectation of the surface is equal to the ground-state of the Hamiltonianassociated with the random-field Gaussian free field. We mention the following question: Whatis the relation, if any, between the ground state of the ∇ φ -model and its thermal expectation? Appendix A. Nash–Aronson estimates for the Dirichlet problem
In this section, we prove the Nash–Aronson estimate in finite volume stated in Proposition 3.3.The proof builds upon the infinite-volume result of [33, Appendix B] stated below (which itselfbuilds upon the infinite-volume and continuous estimate of Aronson [5]). We first introduce theinfinite-volume heat kernel and state the Nash–Aronson estimate for discrete, time-dependentand uniformly elliptic environment of Giacomin–Olla–Spohn [33].
Definition A.1.
Let s ∈ R . For each continuous, time-dependent, uniformly elliptic environ-ment a ∶ [ s , ∞ ) × E ( Z d ) → [ c − , c + ] , each initial time s ∈ [ s , ∞ ) , and each point y ∈ Λ L , weintroduce the infinite-volume heat kernel P a , ∞ to be the solution of the parabolic equation ⎧⎪⎪⎨⎪⎪⎩ ∂ t P a , ∞ ( t, x ; s, y ) − ∇ ⋅ a ∇ P a , ∞ ( t, x ; s, y ) = ( t, x ) ∈ ( s, ∞ ) × Z d ,P a , ∞ ( s, x ; s, y ) = { x = y } x ∈ Z d . The next proposition establishes lower and upper bounds on the map P a , ∞ . Proposition A.2 (Nash–Aronson estimates, Propositions B.3 and B.4 of [33]) . There existsa constant C depending on the dimension d and the ellipticity parameters c − , c + such that, forany pair of times s, t ∈ ( s , ∞ ) with t ≥ s and any pair of points x, y ∈ Z d , (A.1) P a . ∞ ( t, x ; s, y ) ≤ C ∨ ( t − s ) d exp ( − c ∣ x − y ∣ ∨ √ t − s ) . Under the additional assumption ∣ x − y ∣ ≤ √ t − s , one has the lower bound (A.2) P a , ∞ ( t, x ; s, y ) ≥ c ∨ ( t − s ) d . The proof of Proposition A.2 given below relies on analytic arguments. We mention that amore probabilistic approach, relying on the introduction of the random-walk whose generatoris the operator −∇ ⋅ a ∇ (following [33, Section 3.2]) and on stopping time arguments, wouldyield the same result. Proof of Proposition 3.3.
First let us note that by the change of variable ( t − s ) → ( t − s )/ c − , it issufficient to prove the result when c − =
1. Let us fix an integer L ≥
0, let c + ∈ [ , ∞ ) be an elliptic-ity constant, and let s ∈ R . Let a ∶ [ s , ∞ ) × E ( Λ + L ) → [ , c + ] be a continuous time-dependent(uniformly elliptic) environment. For any s ∈ [ s , ∞ ) and any vertex y ∈ Λ L , we denote by P a ( ⋅ , ⋅ ; s, x ) the solution of the parabolic equation (3.3). We extend the environment a to thespace [ s , ∞ ) × E ( Z d ) by setting a ( t, e ) = c + for any pair ( t, e ) ∈ [ s , ∞ ) × ( E ( Z d ) ∖ E ( Λ + L )) ,and let P a , ∞ be the infinite volume heat kernel associated with the extended environment a asdefined in Definition A.1. We prove the upper and lower bounds of Proposition 3.3 separately.We will make use of the notations ∥ P a ( t, ⋅ ; s, y )∥ L ( Λ L ) ∶ = ∑ x ∈ Λ L P a ( t, x ; s, y ) and ∥ P a ( t, x ; s, ⋅ )∥ L ( Λ L ) ∶ = ∑ y ∈ Λ L P a ( t, x ; s, y ) as well as, for any directed edge e = ( x, z ) ∈ ⃗ E ( Λ + L ) , ∇ P a ( t, e ; s, y ) = P a ( t, x ; s, y ) − P a ( t, z ; s, y ) and, following the conventions of Section 2.2, ∥ ∇ P a ( t, ⋅ ; s, y )∥ L ( Λ + L ) ∑ e ∈ E ( Λ + L ) ( ∇ P a ( t, e ; s, y )) . ANDOM-FIELD RANDOM SURFACES 35
Proof of the upper bound.
We first note that the estimate (3.5) is equivalent to the twofollowing inequalities: there exist constants c, C depending on d, c + such that(A.3) ⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩ P a ( t, x ; s, y ) ≤ C ∨ ( t − s ) d exp ( − c ∣ x − y ∣ ∨ √ t − s ) if ( t − s ) ≤ L P a ( t, x ; s, y ) ≤ C ∨ ( t − s ) d exp ( − c ( t − s ) L ) if ( t − s ) ≥ L . We first treat the case ( t − s ) ≤ L . By the maximum principle for the parabolic operator ∂ t − ∇ ⋅ a ∇ , one has the estimate, for any t, s ∈ [ s , ∞ ) with t ≥ s and any x, y ∈ Λ L ,(A.4) P a ( t, x ; s, y ) ≤ P a , ∞ ( t, x ; s, y ) . Combining the inequality (A.4) with Proposition A.2 yields the upper bound, for any s, t ∈ ( s , ∞ ) with t ≥ s and any x, y ∈ Z d ,(A.5) P a ( t, x ; s, y ) ≤ C ∨ ( t − s ) d exp ( − c ∣ x − y ∣ ∨ √ t − s ) . This is (A.3) in the case t ≤ L . We now focus on the case t ≥ L . To this end, we denoteby C Poinc the constant which appears in the Poincar´e inequality, that is, the smallest constantwhich satisfies ∥ u ∥ L ( Λ ℓ ) ≤ C Poinc ℓ ∥ ∇ u ∥ L ( Λ + ℓ ) for any sidelength ℓ ∈ N and any function u ∶ Λ + ℓ → R normalized to be 0 on the boundary ∂ Λ ℓ . We then let c ∶ = / C Poinc , and note thatthis constant depends only on the parameter d .Using that the heat kernel P a solves the parabolic equation (3.3) and the discrete integrationby parts (2.6), we have ∂ t ( e c ( t − s ) L ∥ P a ( t, ⋅ ; s, y )∥ L ( Λ L ) ) = c e c ( t − s ) L L ∥ P a ( t, ⋅ ; s, y )∥ L ( Λ L ) (A.6) − e c ( t − s ) L ∑ e ∈ E ( Λ + L ) a ( t, e ) ( ∇ P a ( t, e ; s, y )) . Using the lower bound a ≥ L (which can be applied since the mapping x ↦ P a ( t, x ; s, y ) is equal to 0 on ∂ Λ L ), we obtain ∑ e ∈ E ( Λ + L ) a ( t, e ) ( ∇ P a ( t, e ; s, y )) ≥ C Poinc L ∥ P a ( t, ⋅ ; s, y )∥ L ( Λ L ) . A combination of (A.6) and (A.7) with the definition of the constant c above yields(A.7) ∂ t ( e c ( t − s ) L ∥ P a ( t, ⋅ ; s, y )∥ L ( Λ L ) ) ≤ − c − e c ( t − s ) L ∥ ∇ P a ( t, ⋅ ; s, y )∥ L ( Λ + L ) By the Nash inequality (see [44]) and the non-negativity of the map P a , there exists a constant C Nash depending only on the dimension d such that(A.8) ∥ P a ( t, ⋅ ; s, y )∥ + d L ( Λ L ) ≤ C Nash ⎛⎝ ∑ x ∈ Λ L P a ( t, x ; s, y )⎞⎠ d ∥ ∇ P a ( t, ⋅ ; s, y )∥ L ( Λ + L ) Applying the inequality (3.4) obtained in Section 3.3, we may simplify the inequality (A.8) andwrite(A.9) ∥ P a ( t, ⋅ ; s, y )∥ + d L ( Λ L ) ≤ C Nash ∥ ∇ P a ( t, ⋅ ; s, y )∥ L ( Λ + L ) . Combining (A.7) and (A.9) yields ∂ t ( e c ( t − s ) L ∥ P a ( t, ⋅ ; s, y )∥ L ( Λ L ) ) ≤ − C Nash e c ( t − s ) L ∥ P a ( t, ⋅ ; s, y )∥ dd + L ( Λ L ) (A.10) ≤ − C Nash ( e c ( t − s ) L ∥ P a ( t, ⋅ ; s, y )∥ L ( Λ L ) ) dd + Integrating the differential inequality (A.10) and using the identity ∥ P a ( s, ⋅ ; s, y )∥ L ( Λ L ) =
1, weobtain that there exists a constant C depending on d and c − such that, for any t ≥ s ,(A.11) e c ( t − s ) L ∥ P a ( t, ⋅ ; s, y )∥ L ( Λ L ) ≤ C ∨ ( t − s ) d . We now show that the estimate (A.11) implies the inequality (A.3) in the case ( t − s ) ≥ L . Usingthe convolution property for the heat kernel P a , we have the identity, for any t, s ∈ [ s + L , ∞ ) and any x, y ∈ Λ L , P a ( t, x ; s, y ) = ∑ z ∈ Λ L P a ( t, x ; t + s , z ) P a ( t + s , z ; s, y ) . The Cauchy-Schwarz inequality then yields(A.12) P a ( t, x ; s, y ) ≤ ∥ P a ( t, ⋅ ; t + s , y )∥ L ( Λ L ) ∥ P a ( t + s , x ; s, ⋅ )∥ L ( Λ L ) . The first term of the right-hand side of (A.12) can be estimated by the upper bound (A.11)applied with the initial time s + t instead of s . We obtain(A.13) ∥ P a ( t, ⋅ ; s + t , y )∥ L ( Λ L ) ≤ Ce − c ( t − s ) L ∨ ( t − s ) d . To estimate the second term in the right-hand side, we use the estimate (A.5), the assumption t − s ≥ L , the observations that, for any pair of points x, z ∈ Λ L , ∣ x − z ∣ ≤ CL , and that thecardinality of the box Λ L is equal to ( L + ) d . We obtain ∥ P a ( t + s , x ; s, ⋅ )∥ L ( Λ L ) ≤ ∑ z ∈ Λ L ⎛⎝ C ∨ ( t − s ) d exp ( − c ∣ x − y ∣ ∨ √ t − s )⎞⎠ (A.14) ≤ ∑ z ∈ Λ L C ∨ ( t − s ) d ≤ CL d ∨ ( t − s ) d ≤ C ∨ ( t − s ) d . A combination of (A.12), (A.13) and (A.14) completes the proof of (A.3) in the case t ≥ L . Proof of the lower bound.
We first claim that there exists a constant c ∈ ( , ) dependingon the parameters d, c + such that, for any t, s ∈ [ s , ∞ ) satisfying √ t − s ≤ c L and any y ∈ Λ L / ,(A.15) sup ( t ′ ,x ) ∈ [ s,t ] × ∂ Λ L P a , ∞ ( t ′ , x ; s, y ) ≤
12 inf x ∈ y + Λ √ t − s P a , ∞ ( t, x ; s, y ) The proof of this inequality relies on Proposition A.2. First by the lower bound (A.2), wehave the estimate, for any t, s ∈ [ s , ∞ ) such that t − s ≤ L ,(A.16) inf x ∈ y + Λ √ t − s P a , ∞ ( t, x ; s, y ) ≥ c ∨ ( t − s ) d ≥ cL d . ANDOM-FIELD RANDOM SURFACES 37
Let us fix a constant c ∈ ( , ) . Using that for any point x ∈ ∂ Λ L and any point y ∈ Λ L / , wehave ∣ x − y ∣ ≥ L / L ≥ c − ,and any t, s ∈ [ s , ∞ ) satisfying √ t − s ≤ c L ,sup ( t ′ ,x ) ∈ [ s,t ] × ∂ Λ L / P a , ∞ ( t ′ , x ; s, y ) ≤ sup ( t ′ ,x ) ∈ [ s,t ] × ∂ Λ L C ∨ ( t ′ − s ) d exp ( − c ∣ x − y ∣ ∨ √ t ′ − s ) (A.17) ≤ sup t ′ ∈ [ s,t ] C ∨ ( t ′ − s ) d exp ⎛⎝ − cL ( ∨ √ t ′ − s ) ⎞⎠ ≤ sup t ′ ∈ [ ,c L ] Ct ′ d / exp ( − cL √ t ′ ) ≤ sup t ′ ∈ [ ,c ] CL d t ′ d / exp ( − c √ t ′ ) . Using that the mapping t ′ ↦ t ′− d / exp ( − c /( √ t ′ )) tends to 0 as t ′ tends to 0, we may select aconstant c ∈ ( , ] such that(A.18) sup t ′ ∈ [ ,c ] Ct ′ d / exp ( − c √ t ′ ) ≤ c , where the constants c, C are the ones which appear in the right-hand sides of (A.16) and (A.17).Let us note that since the constants c, C depend only on the dimension d and the ellipticityconstant c + , the constant c may be chosen so that it depends only on d, c + . Multiplying bothsides of the inequality (A.18) by L − d yields, for any t, s ∈ [ s , ∞ ) such that √ t − s ≤ c L ,sup ( t ′ ,x ) ∈ [ s,t ] × ∂ Λ L / P a , ∞ ( t ′ , x ; s, y ) ≤ sup t ′ ∈ [ ,c ] C L d t ′ d / exp ( − C √ t ′ ) ≤ c L d ≤ inf x ∈ y + Λ √ t − s P a , ∞ ( t, x ; s, y ) . The proof of (A.15) is complete.We now deduce the lower bound (3.6) from the inequality (A.15). To this end, let us fix atime t ∈ ( s, s + c L ) , set ε ∶ = inf x ∈ y + Λ √ t − s P a , ∞ ( t, x ; s, y ) , and define the map P ε a , ∞ ∶ = P a , ∞ − ε .Let us note that the mapping P ε a , ∞ solves the parabolic equation ∂ t P ε a , ∞ ( ⋅ , ⋅ ; s, y ) − ∇ ⋅ a ∇ P ε a , ∞ ( ⋅ , ⋅ ; s, y ) = ( s, t ) × Λ L . By the definition of the parameter ε and the inequality (A.15), the map P ε a , ∞ satisfies theboundary estimates { P ε a , ∞ ( t ′ , x ; s, y ) ≤ ≤ P a ( t ′ , x ; s, y ) ( t ′ , x ) ∈ [ s, t ] × ∂ Λ L ,P ε a , ∞ ( s, x ; s, y ) = { x = y } − ε ≤ { x = y } = P a ( s, x ; s, y ) x ∈ Λ + L . Applying the maximum principle for the parabolic operator ∂ t −∇⋅ a ∇ , we obtain the inequality,for any ( t ′ , x ) ∈ [ s, t ] × Λ L ,(A.19) P ε a , ∞ ( t ′ , x ; s, y ) ≤ P a ( t ′ , x ; s, y ) . Applying the estimate (A.19) at time t ′ = t and using the definition of the parameter ε yields,for any vertex x satisfying ∣ x − y ∣ ≤ √ t − s ,(A.20) 12 P a , ∞ ( t, x ; s, y ) ≤ P ε a ( t, x ; y, s ) ≤ P a ( t, x ; y, s ) . Combining the estimate (A.20) with the lower bound of Proposition A.2 implies P a ( t, x ; y, s ) ≥ c ∨ ( t − s ) d . The proof of the lower bound (3.6) is complete. (cid:3)
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