Strict convexity of the free energy for non-convex gradient models at moderate β
aa r X i v : . [ m a t h - ph ] J a n Strict convexity of the free energy for non-convexgradient models at moderate β Codina Cotar ∗†‡ , Jean-Dominique Deuschel ∗§ , and Stefan M¨uller ∗¶ October 24, 2018
Abstract
We consider a gradient interface model on the lattice with interactionpotential which is a non-convex perturbation of a convex potential. We showusing a one-step multiple scale analysis the strict convexity of the surfacetension at high temperature. This is an extension of Funaki and Spohn’sresult [10], where the strict convexity of potential was crucial in their proof.
AMS 2000 Subject Classification.
Key words and phrases. effective non-convex gradient interface models, surfacetension, strict convexity, Helffer-Sj¨ostrand representation
We consider an an effective model with gradient interaction. The model describesa phase separation in R d +1 , eg. between the liquid and vapor phase. For simplicitywe consider a discrete basis Λ M ⊂ Z d , and continuous height variables x ∈ Λ M −→ φ ( x ) ∈ R . This model ignores overhangs like in Ising models, but gives a good approximationin the vicinity of the phase separation. The distribution of the interface is givenin terms of its Gibbs distribution with nearest neighbor interactions of gradienttype, that is, the interaction between two neighboring sites x, y depends only onthe discrete gradient, ∇ φ ( x, y ) = φ ( y ) − φ ( x ). More precisely, the Hamiltonian isof the form H M ( φ ) = X x,y ∈ Λ M +1 , | x − y | =1 V ( φ ( y ) − φ ( x )) (1.1) ∗ Corresponding Author † Supported by the DFG-Forschergruppe 718 ‘Analysis and stochastics in complex physicalsystems’ ‡ TU Berlin - Fakult¨at II, Institut f¨ur Mathematik, Strasse des 17. Juni 136, D-10623 Berlin,Germany. E-mail: [email protected] § TU Berlin - Fakult¨at II Institut f¨ur Mathematik Strasse des 17. Juni 136 D-10623 Berlin,Germany. E-mail: [email protected] ¶ Max Planck Institute for Mathematics in the Sciences Inselstrasse 22-26 D-04103 Leipzig,Germany. E-mail: [email protected] here V ∈ C ( R ) is a function with quadratic growth at infinity: V ( η ) ≥ A | η | − B, η ∈ R (1.2)for some A > , B ∈ R .For a given boundary condition ψ ∈ R ∂ Λ M , where ∂ Λ M = Λ M +1 \ Λ M , the(finite) Gibbs distribution on R Λ M +1 at inverse temperature β > µ βV M ,ψ ( dφ ) ≡ Z βM,ψ exp( − βH M ( φ )) Y x ∈ Λ M dφ ( x ) Y x ∈ ∂ Λ M δ ψ ( x ) ( dφ ( x )) . Here Z βM,ψ is a normalizing constant given by Z βM,ψ = Z R Λ M +1 exp( − βH M ( φ )) Y x ∈ Λ M dφ ( x ) Y x ∈ ∂ Λ M δ ψ ( x ) ( dφ ( x )) . One is particularly interested in tilted boundary conditions ψ u ( x ) = < x, u > = d X i =1 x i u i for some given ’tilt’ u ∈ R d . This corresponds to an interface in R d +1 which staysnormal to the vector n u = ( u, − ∈ R d +1 .An object of basic relevance in this context is the surface tension or free energydefined by the limit σ ( u ) = lim M →∞ − β log Z βM,ψ u . (1.3)The existence of the above limit follows from a standard sub-additivity argu-ment. In fact the surface tension can also be defined in terms of the partitionfunction on the torus, see below and [10]. In case of strictly convex potential V with c ≤ V ′′ ≤ c (1.4)where 0 < c ≤ c < ∞ , Funaki and Spohn showed in [10] that σ is strictly convex.The simplest strictly convex potential is the quadratic one with V ( η ) = | η | ,which corresponds to a Gaussian model, also called gradient free field or harmoniccrystal. Models with non quadratic potentials V are sometimes called anharmoniccrystals.The strict convexity of the surface tension σ plays a crucial role in the deriva-tion of the hydrodynamical limit of the Landau-Ginsburg model in [10].Under the condition (1.4), a large deviation principle for the rescaled profilewith rate function given in terms of the integrated surface tension has been derivedin [7]. Here also the strict convexity of σ is very important. Both papers [10] and[7] use very explicitely the condition (1.4) in their proof. In particular they rely onthe Brascamp Lieb inequality and on the random walk representation of Helfferand S¨jostrand, which requires a strictly convex potential V .The objective of our work is to prove strict convexity of σ also for some nonconvex potential. One cannot expect strict convexity for any non convex V , see2elow. Our result is perturbative at high temperature (small β ), and shows strictconvexity of σ ( u ) at every u ∈ R for potentials V of the form V ( η ) = V ( η ) + g ( η )where V satisfies (1.4) and g ∈ C ( R ) has a negative bounded second derivativesuch that √ β · k g ′′ k L ( R ) is small enough.Our proof is based on the scale decomposition of the free field as the sum of twoindependent free fields φ and φ , where we choose the variance of φ small enoughto match the non-convexity of g . This particular type of scale decomposition wasused earlier by Haru Pinson in [11], who also suggested to us the use of thisapproach. The partition function Z βN,ψ u can be then expressed in terms of adouble integral, with respect to both φ and φ . We fix φ and perform first theintegration with respect to φ . This yields a new induced Hamiltonian, which isa function of the remaining variable φ . The main point is that our choice of thevariance of φ and smallness of β allow us to show convexity in φ of the inducedHamiltonian. Of course this Hamiltonian is no longer of the simple form (1.1),in particular we lose the locality of the interaction. However an extension of thetechnique introduced in [7] shows strict convexity of σ. The idea behind the proofis that one can gain convexity via integration. This procedure is called ”one stepdecomposition”, since we perform only one integration. Of course this procedurecould be iterated which would allow to lower the temperature. However for generalnon convex g we do not expect that this procedure works at low temperature forevery tilt u .At low temperature an approach in the spirit of [5], [4] looks more promising[1]. Finally note that, due to the gradient interaction, the Hamiltonian has a con-tinuous symmetry. In particular this implies that no infinite Gibbs state existsfor the lower lattice dimensions, d = 1 , M → ∞ ,c.f. [8]. On the other hand, it is very natural in this setting to consider the gradient Gibbs distributions, that is the image of µ V M ,ψ under the gradient op-eration φ ∈ R Z d −→ ∇ φ . It is easy to verify that this distribution depends onlyon ∇ ψ , the gradient of the boundary condition, in fact one can also introducegradient Gibbs distributions in terms of conditional distributions satisfying DLRconditions, c.f. [10]. Using the quadratic bound (1.1), one can easily see thatthe corresponding measures are tight. In particular for each tilt u ∈ R d one canconstruct a translation invariant gradient Gibbs state ˜ µ u on Z d with mean u : E ˜ µ u [ φ ( y ) − φ ( x )] = < y − x, u > . Under (1.4), Funaki and Spohn proved the existence and unicity of extremal,ie. ergodic, gradient Gibbs state, for each tilt u ∈ R . In the case of non convex V ,unicity of the ergodic states can be violated, even at u = 0 tilt, c.f. [2]. Howeverin this situation, the surface tension is not strictly convex at u = 0. We study the convexity properties of the free energy (as a function of the tilt u ) for non-convex gradient models on a lattice. Using the results of [10], we3ork on the torus, instead of the box Λ M , see Remark 2.4 below. Thus, let T dM = ( Z /M Z ) d = Z d mod ( M ) be the lattice torus in Z d , let u ∈ R d and let β >
0. For a function φ : T dM → R , we consider the discrete derivative ∇ i φ ( x ) = φ ( x + e i ) − φ ( x ) (2.5)and the Hamiltonian H ( u, φ ) = X x ∈ T dM d X i =1 (cid:2) V i ( ∇ i φ ( x ) + u i ) + g i ( ∇ i φ ( x ) + u i ) (cid:3) , (2.6)where V i is convex and g i is non-convex (see (2.11) below). We consider thepartition function Z βM ( u ) = Z X e − βH ( u,φ ) m M ( d φ ) , (2.7)where X = { φ : T dM → R : φ (0) = 0 } (2.8)and m M ( d φ ) = Y x ∈ T dM \{ } d φ ( x ) δ ( d φ (0)) , (2.9)and the free energy f βM ( u ) = − β log Z βM ( u ) . (2.10)We will prove Theorem 2.1
Suppose that V i and g i are C functions on R and that there existconstants C , C , C and < C ≤ ( V i ) ′′ ≤ C , − C ≤ ( g i ) ′′ ≤ . (2.11) Set ¯ C = max (cid:18) C C , C C − , (cid:19) . (2.12) If ( g i ) ′′ ∈ L ( R ) and for i ∈ { , , . . . , d } π (12 d ¯ C ) / p βC C || ( g i ) ′′ || L ( R ) ≤ , (2.13) then ( D f βM )( u ) ≥ C | T dM | Id , ∀ u ∈ R d , (2.14) where | T dM | = M d denotes the number of points in T dM . In other words, the freeenergy per particle is uniformly convex, uniformly in M . emark 2.2 The main point is that the convexity estimate (2.14) holds uniformlyin the size M of the torus. Indeed a direct calculation of D f M yields at uD f M ( u ) = (cid:10) D u H ( u, · ) (cid:11) H − var H D u H ( u, · ) , (2.15)where h f i H = R X f ( φ ) e − H ( u,φ ) m M ( d φ ) R X e − H ( u,φ ) m M ( d φ ) (2.16)and var H f = D ( f − h f i H ) E H . (2.17)Now one might expect that a condition like (2.13) implies that (cid:10) ( D u H ( u, · ) (cid:11) H ≥ cC | T dM | Id (see Lemma 4.1 below). The problem is that naively the variance termscales like | T dM | since D u H is a sum of d | T dM | terms. To get a better estimate,one has to show that in a suitable sense, the termscov H ( D u ( V + g )( u + ∇ i φ ( x )) , D u ( V + g )( u + ∇ j φ ( y ))) (2.18)decay if | x − y | is large. If H is not convex such a decay of correlations is, presently,only proved for the class of potentials studied in [6]. As discussed above, theHelffer-Sj¨ostrand estimates do not apply directly. The main idea is to rewrite Z βM ( u ) as an iterated integral in such a way that each integration involves a con-vex hamiltonian to which the Helffer-Sj¨ostrand theory can be applied (see (2.42)below). Remark 2.3
Instead of || g ′′ || L ( R ) one can also use bounds on lower order deriva-tives. More precisely, condition (2.13) can, for example be replaced by50 √ π d ¯ C ( βC ) / C || g ′ || L ( R ) ≤
12 (2.19)(see Remark 4.2 below). In view of the estimate Z R ( g ′ ) ( s ) d s = Z R g ( s ) g ′′ ( s ) d s ≤ C || g || L ( R ) , (2.20)we can see that (2.13) can be replaced by cd ¯ C ( βC ) / C || g || L ( R ) ≤
14 (2.21)with c = π . Remark 2.4
Note that the surface tensions defined in (1.3) and (2.10) coincide(see, for example, [10]). Because of this, we will work from now on with thedefinition of the surface tension on a torus, as it is easier to use.5 utline of the proof for Theorem 2.1
Step 1: Scaling argument
A simple scaling argument shows that it suffices to prove the result for β = 1 , C = 1 . (2.22)Indeed, suppose that the result is true for β = 1 and C = 1. Given β , V i and g i which satisfy (2.11) and (2.13), we define˜ V i ( s ) = βV i (cid:18) s √ βC (cid:19) , ˜ g i ( s ) = βg i (cid:18) s √ βC (cid:19) . (2.23)Then 1 ≤ ( ˜ V i ) ′′ ≤ C C , − C C ≤ ( ˜ g i ) ′′ ≤ , || ( ˜ g i ) ′′ || L ( R ) = p βC C || ( g i ) ′′ || L ( R ) . (2.24)Hence ˜ V i , ˜ g i satisfy the assumptions of Theorem 2.1 with β = 1 and C = 1.Thus D f M ( · , ˜ V i , ˜ g ) ≥ | T dM | Id . (2.25)On the other hand, the change of variables˜ φ ( x ) = p βC φ ( x ) , ˜ u = p βC u (2.26)yields ˜ V i (cid:16) ˜ u i + ∇ i ˜ φ ( x ) (cid:17) = V i ( u + ∇ i φ ( x )) (2.27)and thus Z βM ( u, V i , g i ) = ( βC ) − ( | T dM |− / Z M (˜ u, ˜ V , ˜ g ) . (2.28)Hence f βM ( u, V , g ) = const ( β, C ) + 1 β f M (cid:16)p βC u, ˜ V , ˜ g (cid:17) . (2.29)Thus (2.25) implies (2.13), as claimed. Step 2: Separation of the Gaussian part
Next we separate the Gaussian part in the Hamiltonian. From now on, we willalways assume that β = 1 and C = 1. Set V ( s ) = V ( s ) − s , g = V + g . (2.30)Then 0 ≤ V ′′ ≤ C − , − C ≤ g ′′ ≤ C − H ( u, φ ) = X x ∈ T dM d X i =1
12 ( u i + ∇ i φ ( x )) + G ( u, φ ) , (2.32)where G ( u, φ ) = X x ∈ T dM d X i =1 g ( u i + ∇ i φ ( x )) . (2.33)Since for all functions φ on the torus and for all i ∈ { , , . . . d } X x ∈ T dM ∇ i φ ( x ) = 0 , (2.34)we get H ( u, φ ) = 12 | T dM || u | + 12 ||∇ φ || + G ( u, φ ) , (2.35)where ||∇ φ || = P x ∈ T M P di =1 |∇ i φ ( x ) | . Let Z = Z X e − ||∇ φ || m M ( d φ ) . (2.36)Then the measure µ = 1 Z e − ||∇ φ || m M ( d φ ) (2.37)is a Gaussian measure. Its covariance C is a positive definite symmetric operatoron X (equipped with a standard scalar product ( φ, ψ ) = P x ∈ T dM φ ( x ) ψ ( x )) suchthat ( C − φ, φ ) = ||∇ φ || , ∀ φ ∈ X. (2.38)The partition function thus becomes (recall that we take β = 1) Z M ( u ) = Z e − | T dM || u | Z X e − G ( u,φ ) µ ( d φ ) . (2.39) Step 3: Decomposition of µ and Helffer-Sj¨ostrand calculus By standard Gaussian calculus, µ = µ ∗ µ , where µ and µ are Gaussian withcovariances C = λC, C = (1 − λ ) C, where λ ∈ (0 , . (2.40)More explicitly, for i ∈ { , } µ i ( d φ ) = 1 Z i e − λi ||∇ φ || m M ( d φ ) , where λ = λ, λ = 1 − λ. (2.41)7hus Z M ( u ) = Z e − | T dM || u | Z X Z X e − G ( u,ψ + θ ) µ ( d θ ) µ ( d ψ ) . (2.42)To write the free energy in a more compact form, we introduce the renormalizationmaps R i . For f ∈ C ( R d × X ) we define R i f by e − R i f ( u,a ) := Z X e − f ( u,a + b ) d µ i ( b ) . (2.43)Taking the logarithm of (2.42), we get f M ( u ) = const ( M ) + 12 | T dM || u | + ( R R G )(0 , u ) . (2.44)The main point now is that the map H ( θ ) = G ( u, ψ + θ ) + 12 λ ||∇ θ || (2.45)becomes uniformly convex for sufficiently small λ . This will allow us to use theHelffer-Sj¨ostrand representation to get a good lower bound for D ( R G ), whichinvolves, roughly speaking, the expectation of G i,x ( θ ) = g ′′ ( u i + ∇ i ψ ( x ) + ∇ i θ ( x ))with respect to e − H (see (4.82)). This expectation can be controlled in terms of || g ′′ || L ( R ) (see Lemma 4.1). Under the smallness condition (2.13) one then easilyobtains the lower bound for D ( R R G ) (see (4.85) and (4.87)). Let U and X be finite-dimensional inner product spaces, let C be a positive definitesymmetric operator on X and let µ C be the Gaussian measure with covariance C on X , i.e µ C ( db ) = 1 Z C e − ( C − b,b ) d b, (3.46)where d b is the dim X dimensional Hausdorff measure on X (i.e d b = Q d b i ifthe b i are the coordinates with respect to an orthonormal basis). For a continuousfunction f ∈ C ( U × X ) we define R C f by e − R C f ( u,a ) = Z X e − f ( u,a + b ) d µ C ( d b ) . (3.47)In the situation we will consider, b → f ( u, a + b ) + ( C − b, b ) will be convex andhence bounded from below so that the right hand side of the above identity isstrictly positive.For f ∈ C ( U × X ) we write D f ( u, a ) for the Hessian at ( u, a ), viewed as anoperator from U × X to itself. The restriction of the Hessian to X is denoted by D X f := P X D f P X , where P X is the orthogonal projection U × X → X . On thelevel of quadratic forms we thus have (cid:0) D X f ( u, a )( ˙ u, ˙ a ) , ( ˙ u, ˙ a ) (cid:1) = (cid:0) D f ( u, a )(0 , ˙ a ) , (0 , ˙ a ) (cid:1) . (3.48)8rom the Helffer-Sj¨ostrand representation of the variance (see, e.g., [9] (2.6.15))and the duality relation12 (cid:0) A − a, a (cid:1) = sup b ∈ D ( A ) (cid:18) ( a, b ) −
12 (
Ab, b ) (cid:19) , (3.49)which holds for any positive definite self-adjoint operator A on a Hilbert space Y ,one immediately obtains the following estimate: Lemma 3.1
Supppose that H ∈ C ( X ) , sup X | D H | < ∞ and there exists a δ > such that D H ( a ) ≥ δ Id , ∀ a ∈ X. (3.50) Set Y = { K ∈ L loc ( X ) : (cid:10) | DK | (cid:11) H < ∞} , (3.51) Y = { K ∈ Y : (cid:10) || D K || HS (cid:11) H < ∞} , (3.52) where the derivatives are understood in the weak sense and k DK k HS := X x,y ∈ T dM \{ } (cid:18) ∂ ∂φ ( x ) ∂φ ( y ) K (cid:19) (3.53) denotes the Hilbert-Schmidt norm. Then for all G ∈ Y we have var H G = sup K ∈ Y (cid:10) DG, DK ) − ( DK, D HDK ) − k D K k HS (cid:11) H (3.54) Therefore var H G ≤ sup K ∈ Y (cid:10) DG, DK ) − ( DK, D HDK ) (cid:11) H . (3.55)We will use (3.55) from Lemma 3.1 in the proof of the lemma below. Lemma 3.2
Suppose that f ∈ C ( U × X ) and sup U × X | D f | < ∞ . Supposemoreover that there exists a δ > such that D f ( u, a ) + C − ≥ δ Id , ∀ ( u, a ) ∈ U × X. (3.56) Then Rf ∈ C ( U × X ) and for all u, ˙ u ∈ U , a, ˙ a ∈ X (cid:0) ( D Rf )( u, a )( ˙ u, ˙ a ) , ( ˙ u, ˙ a ) (cid:1) ≥ inf K ∈ Y (cid:10)(cid:0) D f ( u, a + · )( ˙ u, ˙ a − DK ( · ) , ( ˙ u, ˙ a − DK ( · )) (cid:1)(cid:11) H,a + (cid:10) ( C − DK ( · ) , DK ( · )) (cid:11) H u,a (3.57) where H u,a ( b ) = f ( u, a + b ) + 12 ( C − b, b ) , (3.58) h g i H u,a = R g ( b ) e − H u,a ( b ) d b R e − H u,a ( b ) d b . (3.59)9 roof We have e − Rf ( u,a ) = Z X e − [ f ( u,a + b )+ ( C − b,b ) ] d b. (3.60)It follows from (3.56) that f ( u, a + b ) + (cid:0) C − ( a + b ) , ( a + b ) (cid:1) ≥ δ | a + b | − c (3.61)and standard estimates yield f ( u, a + b ) + ( C − b, b ) ≥ δ | b | − c (cid:0) | a | (cid:1) . (3.62)Hence, by the dominated convergence theorem, the right-hand side of (3.60) isa C function in ( u, a ) and the same applies to Rf since the right-hand side of(3.60) does not vanish.To prove the estimate (3.57) for D Rf , we may assume without loss of general-ity that a = 0 , u = 0 (otherwise we can consider the shifted function f ( ·− u, ·− a )).Set h ( t ) := Rf ( t ˙ u, t ˙ a ) . (3.63)Then h ′′ (0) = (cid:0) D ( Rf )(0 , u, ˙ a ) , ( ˙ u, ˙ a ) (cid:1) . (3.64)Now h ( t ) = − log Z X e − f ( t ˙ u,t ˙ a + b ) µ C ( d b ) , (3.65) h ′ ( t ) = R X e − f ( t ˙ u,t ˙ a + b ) Df ( t ˙ u, t ˙ a + b )( ˙ u, ˙ a ) µ C ( d b ) R X e − f ( t ˙ u,t ˙ a + b ) µ C ( d b ) (3.66)and h ′′ (0) = (cid:10)(cid:0) D f (0 , · )( ˙ u, ˙ a ) , ( ˙ u, ˙ a ) (cid:1)(cid:11) H − var H Df (0 , · )( ˙ u, ˙ a ) , (3.67)where H ( b ) = f (0 , b ) + 12 ( C − b, b ) . (3.68)By assumption, D H ( b ) ≥ δ Id , (3.69)i.e. H is uniformly convex. (cid:3) Hence by (3.55) from Lemma 3.1 − var H g ≥ inf K ∈ Y (cid:10) − Dg, DK ) + (
DK, D HDK ) (cid:11) H . (3.70)10pply this with g ( b ) = Df (0 , b )( ˙ u, ˙ a ) (3.71)and write D H = D X f + C − . (3.72)Then − Dg, DK ) + (
DK, D HDK )= − D f (0 , · ) (( ˙ u, ˙ a ) , (0 , DK )) + D f (0 , · ) ((0 , DK ) , (0 , DK ))+( C − DK, DK ) . (3.73)Together with (3.70) and (3.67) this yields (3.57). (cid:3) By (2.44) f M ( u ) = const ( M ) + 12 | T dM || u | + ( R R G )(0 , u ) , (4.74)where G ( u, φ ) = X x ∈ T dM d X i =1 g i ( u i + ∇ i φ ) . (4.75)We first estimate D R G from below. By (2.31)( g i ) ′′ ≥ − C ≥ − ¯ C (4.76)(recall that we always assume C = 1). By (2.12), we have ¯ C ≥
1. If we take λ = 12 ¯ C (4.77)then H u,ψ ( θ ) := G ( u, ψ + θ ) + 1 λ ||∇ θ || (4.78)is uniformly convex, i.e. D H u,ψ ( θ )( ˙ θ, ˙ θ ) ≥ ¯ C ||∇ ˙ θ || ≥ δ M ¯ C || ˙ θ || , (4.79)with δ M >
0. Here we used the discrete Poincare inequality ||∇ η || ≥ δ M || η || for η ∈ X (4.80)11hich follows from a simple compactness argument since T dM is a finite set. Hence,by Lemma 3.2, we have (cid:0) D R ( G )( u, ψ )(¯ u, ¯ ψ ) , (¯ u, ¯ ψ ) (cid:1) ≥ inf K ∈ Y (cid:26) * X x ∈ T dM d X i =1 ( g i ) ′′ ( u i + ∇ i ψ ( x ) + ∇ i · ( x )) (cid:18) u i + ∇ i ψ ( x ) − ∇ i ∂K∂φ ( x ) ( · ) (cid:19) + 1 λ X x ∈ T dM d X i =1 (cid:12)(cid:12)(cid:12)(cid:12) ∇ i ∂K∂φ ( x ) (cid:12)(cid:12)(cid:12)(cid:12) + H u,ψ (cid:27) , (4.81)where Y is defined by (3.52). Now ( g i ) ′′ = ( V i ) ′′ + g ′′ ≥ g ′′ (see (2.30) and(2.31)) and together with the estimate ( a − b ) ≤ a + 2 b and the assumption − C ≤ g ′′ ≤
0, this yields (cid:16) D R ( G )( u, ψ ) , ( ˙ u, ˙ ψ ) , ( ˙ u, ˙ ψ ) (cid:17) ≥ X x ∈ T dm d X i =1 D ( g i ) ′′ ( u i + ∇ i ψ ( x ) + ∇ i · ( x )) ( u i + ∇ i ψ ( x )) E H u,ψ + *(cid:18) λ − C (cid:19) X x ∈ T dM d X i =1 (cid:12)(cid:12)(cid:12)(cid:12) ∇ i ∂K∂φ ( x ) ( · ) (cid:12)(cid:12)(cid:12)(cid:12) + H u,ψ , (4.82)where λ − C ≥
0. We will now use the following result, which will be proven atthe end of this section.
Lemma 4.1
For h ∈ L ( R ) ∩ C ( R ) , ψ ∈ X , x ∈ T dM and i ∈ { , , . . . d } consider F ∈ C ( X ) given by F ( θ ) = h ( u i + ∇ i ψ ( x ) + ∇ i θ ( x )) . (4.83) Then (cid:12)(cid:12)(cid:12) h F i H u,ψ (cid:12)(cid:12)(cid:12) ≤ π (12 d ¯ C ) / || h || L ( R ) . (4.84)Together with (4.82), the smallness condition (2.13) and the relation P x ∈ T dM ∇ i ψ ( x ) =0, this lemma yields D R G ( u, ψ )( ˙ u, ˙ ψ )( ˙ u, ˙ ψ ) ≥ − X x ∈ T dM d X i =1 (cid:12)(cid:12)(cid:12) ˙ u i + ∇ i ˙ ψ ( x ) (cid:12)(cid:12)(cid:12) = − | T dM || ˙ u | − ||∇ ˙ ψ || . (4.85)Thus H ( ψ ) := ( R G )( u, ψ ) + 12(1 − λ ) ||∇ ψ || (4.86)12s uniformly convex and another application of Lemma 3.2 gives (cid:0) D ( R R G )( u, u, , ( ˙ u, (cid:1) ≥ inf K (cid:28) D ( R G )( u, · )( ˙ u, − DK )( ˙ u, − DK ) + 11 − λ ||∇ DK || (cid:29) H ≥ − | T dM || ˙ u | + inf K (cid:26) (cid:18) − λ − (cid:19) (cid:10) ||∇ DK || (cid:11) H (cid:27) ≥ − | T dm || ˙ u | , (4.87)where in the last inequality we used that fact that − λ − ≥
0. In view of (4.74),this finishes the proof of Theorem 2.1.
Proof of Lemma 4.1
Note that u and ψ are fixed. Since the function ˜ h ( s ) = h ( u i + ∇ i ψ ( x ) + s ) has thesame L norm as h , it suffices to prove the estimate for the function F ∈ C ( X )given by F ( θ ) = h ( ∇ i θ ( x )) . (4.88)Moreover, we write H instead of H u,ψ . Letˆ h ( k ) = Z R e − iks h ( s ) d s (4.89)denote the Fourier transform of h . Then || ˆ h || L ∞ ( R ) ≤ || h || L ( R ) (4.90)and h ( s ) = 12 π Z R e iks ˆ h ( s ) d k. (4.91)Set A ( k ) = h F k i H , where F k ( θ ) = e ik ∇ i θ ( x ) . (4.92)Then h F i H = 12 π Z R A ( k ) h ( k ) d k (4.93)and, in view of (4.90), it suffices to show that Z R | A ( k ) | d k ≤ d ¯ C ) / . (4.94)First note that | F k | = 1. Hence | A ( k ) | ≤ , ∀ k ∈ R . (4.95)13o get decay of A ( k ) for large k we use integration by parts. First note that for G i ∈ C ( X ), with sup a ∈ X e − δ | a | ( | G i | ( a ) + | DG i | ( a )) < ∞ for all δ >
0, we have (cid:28) ∂G ∂φ ( x ) G (cid:29) H = (cid:28) − G ∂G ∂φ ( x ) (cid:29) H + (cid:28) ∂H∂φ ( x ) G G (cid:29) H . (4.96)Assume first that x ∈ T dM \ { } . Then F k ( θ ) = − k ∂ F k ∂θ ( x ) ( θ ) (4.97)and thus − k A ( k ) = (cid:28) ∂ F k ∂θ ( x ) · (cid:29) H = (cid:28) ∂F k ∂θ ( x ) ∂H∂θ ( x ) (cid:29) H = − (cid:28) F k ∂ H∂θ ( x ) (cid:29) H + * F k (cid:18) ∂H∂θ ( x ) (cid:19) + H . (4.98)Since | F k | = 1, this yields | A ( k ) | ≤ k (cid:28)(cid:12)(cid:12)(cid:12)(cid:12) ∂ H∂θ ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:29) H + 1 k *(cid:18) ∂H∂θ ( x ) (cid:19) + H . (4.99)Application of (4.96) with G = 1 , G = ∂H∂θ ( x ) gives (cid:28) ∂ H∂θ ( x ) (cid:29) H = *(cid:18) ∂H∂θ ( x ) (cid:19) + H . (4.100)Thus | A ( k ) | ≤ k (cid:28)(cid:12)(cid:12)(cid:12)(cid:12) ∂ H∂θ ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:29) H . (4.101)Now recall that H ( θ ) = X x ∈ T dM d X i =1 g i ( u i + ∇ i ψ ( x ) + ∇ i θ ( x )) + 12 λ |∇ i θ ( x ) | . (4.102)Since λ − = 2 ¯ C , it follows that (cid:12)(cid:12)(cid:12)(cid:12) ∂ H∂θ ( x ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ d (cid:18) sup R (cid:12)(cid:12) ( g i ) ′′ (cid:12)(cid:12) + 1 λ (cid:19) ≤ d ¯ C. (4.103)Hence | A ( k ) | ≤ d ¯ Ck . (4.104)Using (4.104) for | k | ≥ (12 d ¯ C ) / and (4.95) for | k | ≤ (12 d ¯ C ) / , we get (4.94).Finally, if x = 0 we note that F k ( θ ) = − k ∂ ∂θ ( e i ) F k ( θ ) (4.105)and we proceed as before. (cid:3) emark 4.2 The proof shows that for h = g ′′ we can also use norms involvingonly lower derivatives of g . In particular, we have | (cid:10) g ′′ (cid:11) H | ≤ π Z R | ˆ g ′′ ( k ) || A ( k ) | d k ≤ π || ˆ g ′ ( k ) || L ( R ) (cid:18)Z R k | A ( k ) | d k (cid:19) / ≤ √ π || g ′ || L ( R ) (cid:18) (cid:18)
13 + (12 d ¯ C ) (cid:19)(cid:19) / , (4.106)where we used (4.95) for | k | ≤ | k | ≥ Remark 4.3
Note that our proofs can be very easily adapted to any decomposi-tion of µ = µ ∗ µ , where µ and µ are Gaussian with covariances C and C ,such that H u,ψ ( θ ) := G ( u, ψ + θ ) + ( C − θ, θ ) is uniformly convex. Remark 4.4
The procedure for the one-step decomposition can be iterated andthe proofs can be adapted to the multi-scale decomposition; iterating the methodwould lower the temperature and weaken the conditions on the pertubation func-tion g . However, our iteration procedure would not allow us to get results involvingthe low temperature case. Examples (a) V ( s ) = s + a − log( s + a ) , where 0 < a <
1. Then C = C = 2, C = a , || ( g i ) ′′ || L ( R ) = 2 q a and β ≤ a π × d .2468V(s)-4 -2 0 2 4 sFigure 1: Example (a)(b) Let 0 < δ < ( s ) = ( x − δ x ( δ − x ) if 0 ≤ x ≤ δ x otherwise . Then C = C = 1, ¯ C = , || ( g i ) ′′ || L ( R ) ≤ √ δ and β ≤ (cid:16) √ dπ δ (cid:17) .Note that if δ <<
1, the surface tension is convex for very large values of β .0.0020.0040.0060.008V(s)-0.3 -0.2 -0.1 0 0.1 0.2 0.3 sFigure 2: Example (b)(c) Let p ∈ (0 ,
1) and 0 < k < k . Let V ( s ) = − log (cid:18) pe − k s + (1 − p ) e − k s (cid:19) . Then V ′′ ( s ) = pk e − k s + (1 − p ) k e − k s pe − k s + (1 − p ) e − k s and g ′′ ( s ) = − p (1 − p )( k − k ) s p e − ( k − k ) s + 2 p (1 − p ) + (1 − p ) e ( k − k ) s . We have k ≤ V ′′ ( s ) ≤ pk + (1 − p ) k and − p ( k − k )1 − p ≤ g ′′ ( s ) ≤ , where the lower bound inequality for g ′′ ( s ) follows from the fact that g ′′ ( s )attains its minimum for s ≥ q k − k . Then || g ′′ ( s ) || L ( R ) ≤ p − p p ( k − k ) π and β ≤ (cid:18) − p p (cid:19) πk d ¯ C ( k − k ) . V for large enough β . Acknowledgment
Codina Cotar thanks David Brydges and Haru Pinson for invaluable advice andsuggestions during the writing of the manuscript.
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