Decay of covariances, uniqueness of ergodic component and scaling limit for a class of φ systems with non-convex potential

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1 Decay of covariances, uniqueness of ergodic component and scaling limit for a class of φ systems with non-convex potential Codina Cotar and Jean-Dominique Deuschel July 7, 8 Abstract We consider a gradient interface model on the lattice with interaction potential which is a nonconvex perturbation of a convex potential. Using a technique which decouples the neighbouring vertices sites into even and odd vertices, we show for a class of non-convex potentials: the uniqueness of ergodic component for φ- Gibbs measures, the decay of covariances, the scaling limit and the strict convexity of the surface tension. AMS Subject Classification. 6K35, 8B4, 35J15 Key words and phrases. effective non-convex gradient interface models, uniqueness of ergodic component, decay of covariances, scaling limit, surface tension 1 Introduction Phase separation in R d+1 can be described by effective interphase models. In this setting we ignore overhangs and for x Z d, we denote by φx) R the height of the interface above or below the site x. Following a Gibbs formalism, the finite Gibbs distribution ν ψ Λ on RZd of φx)) x Z d ν ψ Λ dφ) = 1 } Z ψ exp { βh ψ Λ φ) dφ Λ, 1) Λ with boundary condition φx) = ψx), if x Λ and normalizing constant Z ψ Λ is characterized by the inverse temperature β > and the Hamiltonian H ψ Λ on Λ, which we assume to be of gradient type: H ψ Λ φ) = V i φx)), ) x Λ where I = { d, d + 1,..., 1,1,,...,d}. 3) and where we introduced for each x Z d and each i I, the discrete gradient i φx) = φx + e i ) φx), Supported by the DFG-Forschergruppe 718 Analysis and stochastics in complex physical systems Corresponding Author TU Berlin - Fakultät II, Institut für Mathematik, Strasse des 17. Juni 136, D-163 Berlin, Germany. cotar@math.tu-berlin.de TU Berlin - Fakultät II Institut für Mathematik Strasse des 17. Juni 136 D-163 Berlin, Germany. deuschel@math.tu-berlin.de

2 that is, the interaction depends only on the differences of neighboring heights. We thus have a massless model with a continuous symmetry! Our state space R Zd being unbounded, such models are facing delocalization in lower dimensions d = 1,, and no infinite Gibbs state exists in these dimensions. Instead of looking at the Gibbs measures of the φx)) x Z d, Funaki and Spohn proposed to consider the distribution of the gradients i φx)),x Z d under ν, the so-called gradient Gibbs measure, which in view of the Hamiltonian ), can also be given in terms of a Dobrushin-Landford- Ruelle description. Assuming strict convexity of V : < C 1 V C < 4) Funaki and Spohn showed in [14], the existence and uniqueness of ergodic gradient Gibbs measures for every tilt u R d, see also Sheffield [1]. Moreover, they also also proved that the corresponding free energy, or surface tension, σ C 1 R d ) is strictly convex. Both results are essential for their derivation of the hydrodynamical limit of the Ginzburg Landau model. In fact under strict the convexity assumption 4) of V, much is known for the gradient field. At large scales it behaves much like the harmonic crystal or gradient free fields which is a gaussian field with quadratic V. In particular Naddaf and Spencer [] showed that the rescaled gradient field converges weakly as ɛ to a continuous homogeneous gaussian field, that is S ɛ f) = ɛ d/ x Z d i φx) u i ) fɛx) N,σ u f)) as ɛ, f C Rd ) see also Giacomin et al. [16] and Biskup and Spohn [4] for similar results). This scaling limit theorem derived at standard scaling ɛ d/, is far from trivial, since, as shown in Delmotte and Deuschel [9], the gradient field has slowly decaying, non absolutely summable covariances, of the algebraic order C cov ν i φx), j φy)) 1 + x y d. 5) The aim of this paper is to relax the strict convexity assumption 4). Our potential is of the form V i φx)) = V i φx)) + g i φx)) where V,g C R) are such that V satisfies 4) and C g, g L 1 R) <, 6) with C > C. Our main result shows that if the inverse temperature β is sufficiently small, that is if: β C 1 g L 1 R) C 1 C d, 7) then the results known in the strick convex case hold. In particular we have uniqueness of the ergodic component at any tilt u R d, strict convexity of the surface tension, scaling limit theorem and decay of covariances. As stated above, the hydrodynamical limit for the corresponding Ginzburg-Landau model, should then essentially follow from these results. Note that uniqueness of the ergodic measures is not true at any β for this type of models: Biskup and Kotecky give an example of non convex V which can be described as the mixture of two gaussians with two different variances, where two ergodic gradient Gibbs measures coexists at u = tilt, cf. Biskup and Kotecky [3] see also Figure 4: Example a) below). At this particular β = β c one expects a non strictly convex free energy Biskup [] personal communication). The situation at lower temperature i.e. large β) is again quite different: using renormalization group techniques, Adams et al. show the strict convexity for small tilt u, cf. [1].

3 In a previous paper with S. Mueller, cf. [8], we have proved strict convexity of the surface tension for moderate β in a regime similar to 5). The method used in [8], based on two scale decomposition of the free field, gives less sharp estimates for the temperature, however it is more general and could be applied to non bipartite graphs. In this paper we use a different technique, which relies on the bipartite property of our model. We consider the distribution of the even gradient that is of φy) φx) where both x,y are even): which is again a gradient field and show that under the condition 5), that the resulting Hamiltonian is strictly convex. The main idea, similar to [8], is that convexity can be gained via integration see also Bracamp et al. [6] for previous use of the even/odd representation). In fact we show more: the Hamiltonian associated to the even variables admits an random walk representation, cf. Helffer and Sjöstrand [17] or Deuschel [11], which is the key tool in deriving covariance estimates such as 7) and scaling limit theorems. The other ingredient is the fact, that given the even gradients, the conditional law of the odd variables is simply a product law. Of course this is a special feature of our bipartite model, in particular it would be quite challanging to iterate the procedure, a scheme which could possibly lower the temperature towards the transition β c. Note that iterating the scheme is an interesting open problem. The rest of the paper is presented as follows: in Section 1 we define the model and recall the definition of gradient Gibbs measures. Section two presents the odd/even characterization of the gradient field, in particular our main result, Theorem 1, shows that the random walk representation holds for the even sites under the condition 5). Section also presents a few examples, in particular we show that our criteria gets very close to the Biskup-Kotecky transition, cf. example.3.. In section 3, we give a proof of the uniqueness of the ergodic component. In view of the product law for conditional distribution of the odd sites given the even gradient, this follows immediately from the uniqueness of the even gradient ergodic measures. Here we adapt the dynamical coupling argument of [14] to our situation. Section 4 deals with the decay of covariances, the proof is based on the random walk representation for the even sites which allows us to use the result of [9]. Section 5 shows the scaling theorem, here again we focus on the even variables and apply the random walk representation idea of [16]. Finally section 6 proves the strict convexity of the surface tension, or free energy, which follows from the convexity of the Hamiltonian for the even gradient. We also show a few useful equalities dealing with the derivative of σ, since they play an important role for the hydrodynamic limits of the Ginzburg Landau model. General Definitions and Notations.1 The Hamiltonian For all x,y Z d, let Let d x y = x i y i, 8) i=1 I = { d, d + 1,..., 1,1,,...,d}. 9) For all i I, we define e i in Z d by e i ) j = δ ij, for all i = 1,,..., d and all j = 1,,...,d 1) and e i = e i, for all i = 1,,..., d. 11) 3

4 For each x Z d and all i I, we denote Let Λ be a finite set in Z d with boundary i φx) = φx + e i ) φx). 1) Λ := {x / Λ, x y = 1 for some y Λ} 13) and with given boundary condition ψ such that φx) = ψx) for x Λ. We consider a gradient interface model with Hamiltonian H ψ Λ φ) = U i i φx)) = [V i i φx)) + g i i φx))], 14) x Λ x+e i Λ Λ x Λ x+e i Λ Λ where φx + e i ) = ψx + e i ), if x Λ and x + e i Λ. For all i I, U i C R) are functions with quadratic growth at infinity: U i η) A η B, η R 15) for some A >,B R. We assume that for all i I, V i,g i C R) satisfy the following conditions: and C 1 V i C, where < C 1 < C 16) C g i, where C > C 1. 17) Remark 1 Note that we can extend the results to the case where we have a perturbation with compact support. More precisely, assume that U i = Y i + h i, where U i satisfies 15), D 1 Y i D and D h i on [a,b] and < h i < D 3 on R \ [a,b], with a,b R and h i a) = h i b) =. Set and g i s) = h i s)1 {s [a,b]} + [ h i b) + h i b)s b)] 1 {s>b} + [ h i a) + h i a)s a)] 1 {s<a} 18) V i s) = Y i s) + h i s)1 {s/ [a,b]} [ h i b) + h i b)s b)] 1 {s>b} [ h i a) + h i a)s a)] 1 {s<a}. 19) Thus, we have V i,g i C R), with D h i s) = g i s) for s [a,b] and g i s) = for s R \ [a,b] and D 1 V i s) = Y i s) + h i s)1 {s/ [a,b]} D + D 3. Note that this procedure can also be extended to the case where h i changes sign more than once.. φ-gibbs Measures Let β >. For a finite region Λ Z d, the Gibbs measure for the field of height variables φ R Λ over Λ is defined by ν ψ Λ dφ) = 1 } Z ψ exp { βh ψ Λ φ) dφ Λ, ) Λ with boundary condition ψ and where } Z ψ Λ R = exp { βh ψ Λ φ) dφ Λ, 1) Λ dφ Λ = x Λ dφx) ) is the Lebesgue measure over R Λ. For A Z d, we shall denote by F A the σ-field genereated of R Zd generated by {φx) : x A}. 4

5 Definition The probability measure ν PR Zd ) is called a Gibbs measure for the φ-field φ-gibbs measure for short), if its conditional probability of F Λ c satisfies the DLR equation for every Λ Z d. ν F Λ c)ψ) = ν ψ Λ ), ν a.e. ψ, It is known that the φ-gibbs measures exist when the dimension d 3, but not for d = 1,, where the field delocalizes as Λ Z d, c.f. [13]. An infinite volume limit thermodynamic limit) for ν ψ Λ and Λ Zd exists only when d 3..3 φ Gibbs Measures.3.1 Notation on Z d Let B Z d := {b = x b,y b ) x b,y b Z d, x b y b = 1,b directed from x b to y b }, 3) B Λ Z d := B Z d Λ Λ), Λ := Λ Λ, 4) B Λ Z d := { } b = x b,y b ) x b Z d \ Λ,y b B Λ Z, x d b y b = 1 5) and { } B Λ Z := b = x d b,y b ) B Z d x b Λ or y b Λ. 6) The height variables φ = {φx);x Z d } on Z d automatically determins a field of height differences φ = { φb);b B Z d}. One can therefore consider the distribution µ of ψ-field under the φ-gibbs measure µ. We shall call µ the ψ-gibbs measure. in fact, it is possible to define the ψ-gibbs measures directly by means of the DLR equations and, in this sence, ψ-gibbs measures exist for all dimensions d 1. A sequence of bonds C = {b 1),b ),...,b n) } is called a chain connecting y and x, x,y Z d, if y b1 = y,x b i) = y b i+1) for 1 i n 1 and x b n) = x. The chain is called a closed loop if x b n) = y b 1). A plaquette is a closed loop A = {b 1),b ),b 3),b 4) } such that {x b i),i = 1,...,4} consists of 4 different points. The field η = {ηb)} R B Z d is said to satisfy the plaquette condition if ηb) = η b) for all b B Z d and b A ηb) = for all plaquettes in Z d, 7) where b denotes the reversed bond of b. Let χ be the set of all η R B Z d which satisfy the plaquette condition and let L r,r > be the set of all η RB Z d such that η r := ηb) e r xb <. 8) b B Z d We denote χ r = χ L r equipped with the norm r. For φ = φx)) x Z d and b B Z d, we define the height differences η φ b) := φb) = φy b ) φx b ). 9) 5

6 Then φ = { φb)} satisfies the plaquette condition. Conversely, the heights φ η,φ) R Zd can be constructed from height differences η and the height variable φ) at x = as φ η,φ) x) := b C,x ηb) + φ), 3) where C,x is an arbitrary chain connecting and x. Note that φ η,φ) is well-defined if η = {ηb)} χ..3. Definition of φ-gibbs measures We next define the finite volume φ-gibbs measures. For every ξ χ and Λ Z d the space of all possible configurations of height differences on B Λ Z d for given boundary condition ξ is defined as χ B Λ Z d,ξ = {η = ηb)) b B Λ Z d ;η ξ χ}, 31) where η ξ χ is determined by η ξ)b) = ηb) for b B Λ and = ξb) for b B Λ. The finite Z d Z d volume φ-gibbs measure in Λ or more precisely, in B Λ ) with boundary condition ξ is defined by Z d µ ξ Λ dη) = 1 exp Z λ.ξ β U i ηb)) dη Λ,ξ Pχ B Λ 3) Zd,ξ), b B Λ Z d where dη Λ,ξ denotes a uniform measure on the affine space χ and Z B Λ Z d,ξ Λ,ξ is the normalization constant. Let Pχ) be the set of all probability measures on χ and let P χ) be those µ Pχ) satisfying E µ [ ηb) ] < for each b B Z d. For every ξ χ and a R, let ψ = φ ξ,a be defined by 3) and consider the measure ν ψ Λ. Then µ ξ Λ is the image measure of νψ Λ under the map {φx)} x Λ {ηb) := φ ψ)b)}, b B Λ Z d. 33) Note that the image measure is determined only by ξ and is independent of the choice of a. Definition 3 The probability measure µ Pχ) is called a Gibbs measure for the height differences φ-gibbs measure for short), if it satisfies the DLR equation µ F BZ )ξ) = µ ξ d \B Λ Z d Λ ), µ a.e. ξ, for every Λ Z d, where F BZ d \B Λ Z d stands for the σ-field of χ generated by {ηb);b B Z d \ B Λ Z d }. We will define by GH) := {µ P χ) : µ shift invariant φ Gibbs measure such that 3 Even/Odd Representation 3.1 Notation on the Even Subset of Z d µ ξ Λ has Hamiltonian Hξ Λ }. 34) As Z d is a bipartite graph, we will label the vertices of Z d as even and odd vertices, such that every even vertex has only odd nearest neighbor vertices and vice-versa. Let E d := { a = a 1,a,...,a d ) Z d 6 d i=1 } a i = p,p Z, 35)

7 Figure 1: The bonds of in E O d := { a = a 1,a,...,a d ) Z d d i=1 } a i = p + 1,p Z, 36) E d Λ := E d Λ, O d Λ := O d Λ and O d Λ := O d Λ. 37) We will next define the bonds in E d in a similar fashion to the definitions for bonds on Z d. For all x,y Z d, let } B E d := {b = x b,y b ) x b,y b E d, x b y b =,b directed from x b to y b, 38) B Λ E d := B E d Λ Λ), B Λ E d := { } b = x b,y b ) x b E d \ Λ,y b EΛ, d x b y b = 39) and E d Λ := {y E d Λ y = y b for some b B Λ E d }. 4) An even plaquette is a closed loop A E = {b 1),b ),...,b n) }, where n {3,4}, such that {x b i),i = 1,...,n} consists of n different points. The field η = {ηb)} R B E d is said to satisfy the even plaquette condition if ηb) = η b) for all b B E d and ηb) = for all even plaquettes in E d. 41) b A E Let χ E be the set of all η R B E d which satisfy the even plaquette condition. For each b B E d we define the even height differences η E b) := E φb) = φy b ) φx b ). 4) The heights φ ηe,φ) can be constructed from the height differences η E and the height variable φ) at a = as φ ηe,φ) a) := η E b) + φ), 43) b C E,a where a E d and C E,a is an arbitrary path in Ed connecting and a. Note that φ η,φ) a) is welldefined if η E = {η E b)} χ E. We also define χ E r similarly as we define χ r. As on Z d, let Pχ E ) be the 7

8 set of all probability measures on χ E and let P χ E ) be those µ Pχ E ) satisfying E µ [ η E b) ] < for each b B E d. We will define by G E H) := {µ P χ E ) : µ shift invariant φ Gibbs measure such that for all Λ Z d, µ ξ E d Λ has Hamiltonian H ξ E d Λ}. 44) 3. Restriction of a φ-gibbs measure to E d For simplicity of calculations, we will assume for the remainder of the paper that for all i I, U i = U i and that U i are symmetric, though this condition is not necessary. Note that throught the paper we will use the notation Let i φx) = φx + e i ) φx). 45) θx) = φx + e 1 ),...,φx + e d ),φx e 1 ),...,φx e d )). 46) Lemma 4 Let µ GH), where H has the form in 14). Then µ E d G E H ) ), where H ) φ) := F EΛ d x θx)), 47) x O d Λ where Proof. Let F x θx)) = log e β U i i φx)) dφx). 48) R F Z d := σ φx)),x Z d) and F E d := σ φx),x E d) 49) and F BZ d := σ ηb),b B Z d) and F B E d := σ η E b),b B E d). 5) Since µ GH), for all Λ finite sets in Z d and for all A F BZ we have d µa F BZ )ξ) = µ d \B Λ Λ,ξ A) = 1 e βhξ Λ φ) dφx) Z d Z Λ,ξ A x Λ y Z d \Λ δφy) ξy)), 51) where Z Λ,ξ is the normalizing constant. Let Λ E be a finite set in E d. We will construct from Λ E a finite set Λ Z d as follows: if x Λ E, then x+e i Λ for all i I, Λ E d = Λ E and such that Λ and Λ E have the same boundary conditions, that is only even vertices are boundary points see Figures and 3). Then, since µ is a φ-gibbs measure and since for every Λ Z d, F F BE d \B ΛE B, Z d \B Λ Z it follows that for every A F d BZ d µa F )ξ) = E BE d µ E \B ΛE µ 1 A F BZ d \B Λ = E µ e βh Λφ) dφx) A x Λ Z d ) F BE d \B ΛE )ξ) = E µ µ Λ,ξ A) ) FBE ξ) d \B ΛE δφy) ξy)) FBE ξ), 5) d \B ΛE y Z d \Λ 8

9 where F = σ ηb),b B BE d \B ΛE Ed \ B Λ E). In particular, from 5), by integrating out the odd points and due to the boundary conditions on Λ, we have for every A F B Λ E d µa F )ξ) BE d \B ΛE 1 = e x O Λ d Fx Λ φx+e 1 ),...,φx e d )) Z Λ,ξ A x O d Λ : x+e i E d Λ dφx + e i ) y E d \Λ E δφy) ξy)) 53) Therefore µ E d satisfies the DLR equations, so the proof is finished. Remark 5 Note that for any constant C d = C,C,...,C) R d, we have In particular, this means that for any i I F x θx)) = F x θx) + C d ). 54) F x θx)) = F x φx + e 1 ) φx + e i ),...,φx e d ) φx + e i )). 55) Therefore we are still dealing with a gradient system, even though this is no longer a two-body gradient system. Remark 6 Note that the new Hamiltonian H ) depends on β through the functions F x. Lemma 7 The conditional law of φx)) x O d given F E d is just the product law with density at x O d µ x dφx) F ) = 1 BE d Zθx)) exp β ) U i φx + e i ) φx)) dφx) 56) with Zθx)) = exp β ) U i φx + e i ) φx)) dφx). 57) Proof. First note that for Λ = x O d, 5) becomes µ F BZ d \B x Z d)ξ) = µξ x. 58) Figure : The graph of Λ E Figure 3: The graph of Λ 9

10 Note now that because F BE d x O df B Z d \B x Zd, we can reason as in 5) to get µ F BE d )ξ) = x O dµ F B Z d \B x Z d)ξ) = x O dµξ x. 59) The statement of the lemma follows now from 59). As an immediate consequence of Lemma 7 and of Remark 5, we have Corollary 8 The conditional law of φx)) x O d given F E d is just the product law with density at x O d µ x dφx + e k ) dφx) F ) BE d 1 = Z θx)) exp β ) U i φx) φx + e k ) φx + e i ))) dφx), 6) which depends only on the even gradients E φ, where we defined Z θx)) := Zφx + e 1 ) φx + e k ),...,φx e d ) φx + e k )), 61) 3.3 Random Walk Representation Definition and Theorems Definition 9 Let x O d. We say that F x satisfies the random walk representation, if there exists c 1,c > such that for all i,j I D i,i F x = D i,j F x and c 1 D i,j F x c for i j, 6) where for i I, we denoted by j:j i D i Fy 1,...,y d,y 1,...,y d ) := y i Fy 1,... y d,y 1,...,y d ). 63) The main result of this section is: Theorem 1 Random Walk Representation) For all i I, let U i C R) be such that they satisfy 15). We also assume that, for all i I, V i,g i C R) satisfy 16) and 17). Then, if β sup g i C L 1 R) < C 1, 64) 1 C d there exists c 1,c > such that for all x O d Λ, F x satisfies the random walk representation. Lemma 11 Suppose x O d Λ. Then for all j I such that x + e j EΛ d, we have D j F x θx)) = D i F x θx)), 65) D j,j F x θx)) = and for all i I,i j such that x + e i E d Λ,i j,i j D i,j F x θx)), 66) D i,j F x θx)) = β cov µx U i i φx)),u j j φx)) ), 67) where µ x is given by 56) and E µx and cov µx are the expectation, respectively the covariance, with respect to the measure µ x. 1

11 Proof. For all j I, from 55) we have and for i j D j F x θx)) = φx + e j ) F xφx + e 1 ) φx + e j ),...,φx e d ) φx + e j )) = D i F x φx + e 1 ) φx + e j ),...,φx e d ) φx + e j )) 68) D i F x θx)) =,i J It follows now from 68) and 69) that φx + e i ) F xφx + e 1 ) φx + e j ),...,φx e d ) φx + e j )) = D i F x φx + e 1 ) φx + e j ),...,φx e d ) φx + e j )). 69) D j F x θx)) =,i j By simple differentiation in F x, we have for for all i,j I,i j D i F x θx)). 7) D i,j F x θx)) = β cov µx U i i φx)),u j jφx)) ). 71) 66) now follows immediately from 7) and 71). The following lemma is elementary to prove by using Taylor expansion and will be needed for the proof of Theorem 1: Lemma 1 Representation of Covariances) Let k N. For all functions F,G C 1 R k ; R) and for all measures µ PR k ), we have cov µ F,G) = 1 [Fφ) Fψ)] [Gφ) Gψ)] µdφ)µdψ) = 1 [φ ψ) DFφ,ψ)] [φ ψ) DGφ,ψ)] µdφ)µdψ) where we denoted by 7) and by DFφ,ψ) := 1 DF ψ + tφ ψ)) dt, DGφ,ψ) := DFφ) := 1 DGψ + sφ ψ)) ds 73) ) D 1 Fφ),...,D k Fφ). 74) Proof of Theorem 1 It follows from Lemma 11 that, in order to be able to use the random walk representation, all we need is to estimate the covariances: cov µx U i i φx)),u j j φx)) ). We have U i = V i + g i and U j = V j + g j, where C 1 V i, we see that V j,v j C. Then using Lemma 1 for V i and C 1var µx φx)) C 1 cov µx φx),v j j φx))) cov µx V i i φx)),v j j φx))) C cov µx φx),v j j φx))). 75) 11

12 Since g i,g j < and since we have cov µx g i i φx)),g j j φx))) = cov µx g i i φx)), g j j φx))), 76) we can also estimate cov µx g i iφx)),g j jφx))) using Lemma 1 to get cov µx g i iφx)),g j jφx) C var µ x φx)). 77) We now have to find an upper bound for var µx φx)). From 75), it is sufficient to find an upper bound for cov µx φx),v j jφx)). Note now that from 75), we have cov µx φx),v j jφx) 1 dc 1 cov µx V j jφx)), V i iφx)) ). 78) Using integration by parts, we have cov µx V j jφx)), ) V i iφx) = 1 β E µ x V j jφx)) ) cov µx V j jφx)), g i iφx)) ). 79) It now remains to estimate cov µx V i iφx)),g j jφx)). By using Lemma 1, we get that cov µx V j jφx)), i g i iφx)) < dc cov µx φx),v j jφx))), 8) which has the wrong sign. But by using the Cauchy-Schwarz inequality and 75), we have cov µx V i iφx)),g j jφx)) var µx V i iφx)) var µx g j jφx)) C cov µx φx),v i iφx))) var µx g j jφx)). 81) Then we estimate var µx g j jφx)) by applying Lemma 1 to get var µx g j j φx)) = 1 [ 1 φx) ψx)) = 1 [ φx) φx+ej ) ψx) φx+e j ) g g j s) ds From 78), 79) and 8), we now get the upper bound cov µx φx),v j jφx))) C dβc 1 + j ψx) + tφx) ψx)) φx + e j)) dt] µ x dφ)µ x dψ) ] µ x dφ)µ x dψ) 1 g j L 1 R). 8) C sup g i C L 1 1 R) cov µx φx),v j jφx))), 83) from which we get cov µx φx),v j j φx))) C sup g i L C 1 R) C C 1 ) sup g i L 1 R) + C. 84) dβc 1 1

13 Also, by using 75), we get from 84) var µx φx)) 1 C 1 C C 1 sup g i L 1 R) + 1 C C 1 ) sup g i L 1 R) + C := σ. 85) dβc 1 The upper bound now follows from 75), 77), 84) and 85). To find a lower bound, note now that from 75) we get cov µx φx),v j jφx))) 1 cov µx V j dc jφx)), ) V i iφx)). 86) By using 79) and 8), we get From 75), 77) and 8), we get cov µx U i i φx)),u j j φx)) ) cov µx φx),v j jφx))) [C 1 cov µx φx),v The lower bound now follows from 87) and 88). cov µx φx),v j j φx))) C 1 dc β. 87) j jφx))) C / sup ] g i L 1 R). Remark 13 Note that condition 64) can be replaced by other conditions. For example a) β) 1/4 sup g i L R) < C 1) 3/ C ) 3/4 d 88) 1/4. 89) We obtain this condition by using the Cauchy-Schwartz inequality and 75) covµx V i i φx)),g j j φx)) var µx V i iφx)) var Hx g j jφx)) C cov µx φx),v i iφx))) var µx g j jφx)). 9) Then we estimate var µx g j jφx)) by applying Lemma 1 and Jensen s inequality to get var µx g j j φx)) = 1 [ 1 φx) ψx)) 1 1 φx) ψx)) = 1 φx) φx+ej ) φx) ψx)) ψx) φx+e j ) 1 g j L R) φx) ψx) µ µx dφ)µ µx dψ) g j ψx) + tφx) ψx)) φx + e j)) dt] µ µx dφ)µ µx dψ) [ g j ψx) + tφx) ψx)) φx + e j )) ] dt µµx dφ)µ µx dψ) [ g j s) ] ds µµx dφ)µ µx dψ) 1 g j L R) φx) ψx)) µ µx dφ)µ µx dψ) = 1 g j L R) var Hx φx)) g j cov µx φx)),v i iφx)) L R). 91) C 1 13

14 The rest of the argument to obtain the bound in 64) follows the same steps as in proof of Theorem 1. b) Another possible condition is β) 3/4 sup g i L R) C 1) 3/ 1 C ) 5/4, 9) d) 3/4 obtained by using the Cauchy-Schwarz inequality, 75) and Lemma 14 below, to get cov µx V i iφx)),g j jφx)) var µx V i iφx)) var µx g j jφx)) Lemma 14 If h L 1 R), then we have C ) 3/4 dβ) 1/4 cov µx φx)),v i iφx)))sup g i ) L 1 R). 93) E µx h) dβc h L 1 R). 94) Proof. Using integration by parts and Cauchy-Schwartz, we have E µx h) = E y µ x hz)dz)) = y E y µ x H xy) hz)dz)) E 1/ µ x H x ) ) y ) ) y E 1/ µ x hz) dz = E 1/ µ x H x )E1/ µ x hz) dz dβc h L 1 R). 95) ) Note that we also used property 15) and integration by parts in the above formula. Remark 15 As we mentioned before, we can adapt all the reasoning in Section 3. and Section 3.3 to the case where U i do not fullfil the assumption that for all i I, U i = U i and U i symmetric. In this more general case, F x θx)) = log e β [U i φx))+u i φx))] dφx). 96) R Remark 16 Note that if we consider the case where for all i I, U i are strictly convex such that C 1 U i C, in view of 75), 87) and 83) applied to the case with g = 3.3. Examples C 1 dβc cov µx U i i φx)),u j j φx)) ) C dβc 1. 97) a) Let p,1) and < k < k 1. Let Us) = log ) pe k 1 s + 1 p)e k s. Set a = k 1 k. Take p < a 1 in order that the potential U is non-convex. If < β) 3/4 p1 p) 1/4 a 1) 1/ d) 3/4 π) 1/4, 98)

15 Us) s Figure 4: Example a) then 9) is satisfied and the RW representation holds. If β = 1 and k 1 k, the above a condition is equivalent to p < p, where p 1/4. This is close to, for d =, the d) 3/4 π 1/4 p critical point p c, such that c 1 p c = a 1/4, of [3], where uniqueness of ergodic states is violated for this example of potential U. The computations follow. Take and V s) = pk 1e k s p)k e k s pe k 1 s + 1 p)e k s g p1 p)k 1 k ) s s) = p e k 1 k ) s + p1 p) + 1 p) e k 1 k ) s. We have k V s) pk p)k and pk 1 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 k 1 k. Then g s) L R) p ) p 1 p k 1 k ) 1/4 π) 1/4 + o 1 p k 1 k ) 1/4 π) 1/4. By using condition 9), the RW representation holds. b) Us) = s + a logs + a), where < a < 1. Let < β < a 4d+ 5a). Then the RW representation holds. Then, using the notation from Remark 1, take Y s) = s and hs) = logs + a). We have Y s) =, so D 1 = D = ; also h s) = s a, with s +a) a h s) for s [ a, a] and < h s) 5a otherwise. Then C = a, C 1 =,C = + 5a and g s) L 1 R) = a. By using condition 64), the RW representation holds. 15

16 Vs) s Figure 5: Example b) Us) s Figure 6: Example c) c) Let < a < 1 and Us) = { s s + s a + logs + a) if s + loga s) if s <. Then { s + U 1 s) = s+a if s s + a + 1 s a if s < and U s) = { 1 1 s+a) if s 1 1 s a) if s <. U is non-convex on the interval a 1 < s < 1 a and achieves its unique minimum on < s < a 1. Let < β < a 4. Then the RW representation holds. d Then take V s) = s / and gs) = logs+a) if s and gs) = s/a+loga s) if s <. We have V s) = 1, so C 1 = C = 1; also g s) = 1/s + a) if s and g s) = 1/s a) for s < ; also g s) L 1 R) = a. By using condition 64), the RW representation holds. Note that by using 89), the condition on β becomes < β < 3a3 d becomes < β a1/3 1/3. d and by using 9), it 16

17 4 Uniqueness of ergodic component In this section, we extend to a class of non-convex potentials, the uniqueness of ergodic component result, proved for strictly convex potentials in [14]. We denote by S the class of all shift invariant µ P χ) which are stationary and by ext S those µ S which are ergodic with respect to shifts. For each u R d, we denote by ext S)ũ the family of all µ ext S such that E µ ηe α )) = u α,α = 1,... d, where we denoted by e α the bond e α,). We will prove that Theorem 17 Let U i = V i +g i, where U i satisfy 15) and V i and g i satisfy 16), 17) and 64). Then for every u R d, there exists at most one ergodic µ GH) such that E µ η t e α )) = u α,α = 1,... d. The proof will be done in steps: first, we will prove the uniqueness of ergodic component on E d and then we will use this result combined with the properties of the φ-gibbs measure to extend the result to µ. 4.1 Uniqueness of ergodic component for the even Let F C R d ; R) be such that for all a 1,a,...,a d,a 1 a d ) R d and for all c R Fa 1,...,a d,a 1,... a d ) = Fa 1 + c,...,a d + c,a 1 + c,...,a d + c). 99) Note that from property 99), by the same reasoning as in Lemma 11 we have that for all j I, 65) and 66) hold. Assume that there exist c > and c + > such that for all a 1,a,...,a d,a 1 a d ) R d c D i,j Fa1,a,...,ad,a 1a d) c+. 1) Let L = { } F C R d ; R) F satisfies 99) and 1). 11) The proofs in this section follow very closely the arguments from [14]. To make the current paper self-contained, we will sketch proofs for all the theorems in the section. There are three main ingredients necessary in proving uniqueness of ergodic component for a Hamiltonian satisfying 99) and 1). These steps are: the study of the dynamics of the height variables, which dynamics are generated by SDE, a coupling argument and the use of the ergodicity Dynamics Suppose the dynamics of the even height variables φ t = {φ t a)} R Ed are generated by the SDE dφ t a) = φa) F xφ t x + e 1 ),...,φ t x + e d ))dt + dw t a), 1) x O d Λ, x a =1 where for all x O d Λ, F x L and {W t a),a E d } is a family of independent Brownian motions. Note that in 1), for each x O d Λ such that x a = 1, there exists i I such that a = x + e i. The dynamics for the even height differences η E = {ηt E b)} B E d are determined by the SDE dη E b) = dφ t x b ) dφ t y b ) = + x O d Λ, x y b =1 x O d Λ, x x b =1 φy b ) F xφ t x + e 1 ),...,φ t x + e d ))dt φx b ) F xφ t x + e 1 ),...,φ t x + e d ))dt + d[w t x b ) W t y b )], 13) 17

18 where b = x b,y b ) B Λ E d. Lemma 18 a) The solution of 13) satisfies η E t χ E for all t >. b) If φ t is a solution of 1), then η E t := E φ t is a solution of 13). c) If η E t is a solution of 13) and we define φ t ) through 1) for x = and E φ t b) = η E t b), with φ ) R, then φ t := φ ηe t,φt) is a solution of 1). d) For each η E χ E r,r > the SDE 13) has a unique χe r -valued continuous solution ηe t starting at η E = ηe. Proof. The proofs for a), b) and c) are immediate, so we will concentrate on the proof for d). For every θ t x) and θ t x), by expanding D j F x θ t x)) in Taylor series around θ t x) to get D j F x θ t x)) D j F x θ t x)) = k I φ t x + e k ) 1 D j,k F x θt x) + y θ t x) θ t x) )) dy. 14) By using now the fact that F x F, we obtain global Lipschitz continuity in χ E r of the drift term of the SDE in 13), from which existence and uniqueness of the solution in 13) follows. First, we will prove Lemma 19 Let φ t and φ t be two solutions for 1) and set φ t a) := φ t a) φ t a), where a E d. Then for every Λ Z d, we have t φ t a)) = [ D j F x φ t x + e 1 ),...,φ t x + e d )) a E d Λ x O d Λ {j I x+e j E d Λ } ] D j F x φ t x + e 1 ),..., φ t x + e d )) φ t x + e j ), 15) where t φ t a)) c a E d Λ b B Λ E d [ ] E φt b) + 4c+ φ t y b ) E φt b). 16) b B Λ E d Proof. From 1), we have t φ t a)) = x O d Λ, x a =1 [ φa) F xφ t x + e 1 ),...,φ t x + e d )) ] φa) F x φ t x + e 1 ),..., φ t x + e d )) φ t a). 17) By summing now over all a EΛ d in 17), we get 15). For simplicity of notation, we will denote as before by θ t x) := φ t x + e 1 ),...,φ t x + e d )) and by θ t x) := φt x + e 1 ),..., φ t x + e d ) ). To find an upper bound for S 1, we expand now D j F x θ t x)) in Taylor series around θ t x) to get D j F x θ t x)) D j F x θ t x)) = k I φ t x + e k ) 1 D j,k F x θt x) + y θ t x) θ t x) )) dy. 18) 18

19 By using now 1), 18) and the equality D k,j F x = D j,k F x in 15), we obtain t φ t a)) a E d Λ = x O d Λ = x O d Λ = x O d Λ + x O d Λ c {j I, x+e j E d Λ } {j I, x+e j E d Λ } φ t x + e k ) φ t x + e j ) k I {j,k I,j k x+e j,x+e k E d Λ } {j I, x+e j E d Λ } b B Λ E d 4.1. Coupling Argument We will call k I,k j 1 1 [ φt x + ej ) φ t x + e k ) φ t x + e j )] D j,k F x yφt x) + 1 y) φ t x) ) dy D j,k F x φt x) + y φ t x) φ t x) )) dy [ φt x + e j ) φ 1 t x + e k )] D j,k F x yφt x) + 1 y) φ t x) ) dy {k I x+e j E Λ d } 1 [ φt x + ej ) φ t x + e k ) φ t x + e j )] D j,k F x yφt x) + 1 y) φ t x) ) dy [ ] E φt b) + 4c+ φ t y b ) E φt b). 19) b B Λ E d the set of neighbours of in E d. Let N = {f ij f ij = e i + e j, where i,j I,j i,i j} 11) N + = {f ij f ij = e i + e j, where i,j I,j i,i j,j 1} 111) Let us define now a generator set in E d : and e E 1 = e 1 + e,...,e E d 1 = e d 1 + e d,e E d = e d e 1 if d even 11) e E 1 = e 1 + e,...,e E d 1 = e d 1 + e d,e E d = e d + e 1 if d odd. 113) For each u R d, let ũ α,α = 1,,...,d be defined as follows: and ũ α = u α + u α+1, α = 1,,...,d 1 and ũ d = u d u 1 if d even 114) ũ α = u α + u α+1, α = 1,,...,d 1 and ũ d = u d + u 1 if d odd. 115) 19

20 For x B E d, we define the even shift operators σx E : REd R Ed, for even heights by σx Ey) = φy x) for y E d and φ R Ed and for even height differences by σx E η)b) = ηb x), for b B E d and η χ E. We denote by S E the class of all shift invariant with respect to the even shifts) µ P χ E ) which are stationary for the SDE 1) and by ext S E those µ E S E which are ergodic with respect to the even shifts. For each u R d, we denote by ext S E) the family of all ũ µ E ext S E such that E µ E η E t e E α) ) = ũ α,α = 1,... d. We will prove that µ is unique. Next, for clarity purposes, we will sketch the coupling argument used in [14] to prove uniqueness of µ. Suppose that there exist µ E ext S E) ũ and µe ext S E) ṽ for u,v Rd. Let us construct two independent-χ E r valued random variables η E = {η E b)} and η E = { η E b)} on a common probability space Ω,F,P) in such a manner that η E and η E are distributed by µ E and µ E respectively. We define φ = φ η, and φ = φ η,. Let φ t and φ t be two solutions of the SDE 1) with common Brownian motions having initial data φ and φ. Since µ E, µ E S E, we conclude that ηt E = η E) φ t and η E = η E) φ t are distributed by µ E and µ E respectively, for all t. Let ũ and ṽ be such that ũ α = E µ E η E t e E α) ) and ṽ α = E µ E η E t e E α) ). We claim that: Lemma There exists a constant C > independent of ũ,ṽ R d such that lim 1 T d [ T E P ηt E T ee α ) η ) ] t EeE α ) dt C ũ ṽ. 116) α=1 Proof. Step 1. For simplicity of notation, we will label for this proof the d elements of N + as f1,f,...,f d. By applying Lemma 19 to the differences { φ t x) := φ t x) φ t x)}, where x E d, one obtains just as in [14] for every T >, Λ = Λ N, where N N T gt)dt where c + = 4c + + C, d c B Λ E d EP gt) = d α=1 x E d Λ φ x)) + c +c ) d c N) T E P [ φ t f α )) ] and c := sup {N 1} { sup φ t y) L P) dt, 117) y EΛ d N BΛ E d B Λ E d } <. 118) Step. Next we derive, just as in [14], the following bound on the boundary term: For each ɛ > there exists an l N such that t ) sup φ t y) L P) C 1 ɛ N + N ũ ṽ + N t gs) ds 119) y EΛ d for every t > and l l, where C 1 > is a constant independent of ɛ,l, and t. To this end, because E d is a subalgebra, we can use the mean ergodic theorem for cocycles see for example [5], [19] or [18]) and apply it to µ E ext S E ) u to obtain lim x, x E d 1 x φν, x) x ũ L µ E ) = 1) In order to apply the proof from Step in [14], we will need the result proved below in 13). By using 15) and the reasoning used to derive 7), we have { } φt a) = [D j F x θ t x)) D j F x θt x) ) ] t a Λ = x Λ O d {j I x+e j E d Λ } {x Λ O d,j I x+e j E d Λ,i j such that x+e i / Λ } [ D j F x θ t x)) D j F x θt x) ) ], 11)

21 where Λ = Λ [N/] EΛ d, θ tx + e i ) = φ t x + e i ),...,φ t x + e i )) R d and θ t x + e i ) = φ t x + e i ),..., φ t x + e i ). By using Taylor s expansion and 1) in 11), we get D j F x θ t x)) D j F x θt x) ) c + φx + e k ) φx + e j ). 1) k I,k j Then D j F x θ t x) θ t x + e i )) D j F x θt x) θ t x + e i ) ) L P) {x Λ O d,j I x+e j E d Λ,i j such that x+e i / Λ } c + B Λ d φ t fα ) L P), 13) d α=1 where we define B Λ := {b = x b,y b ) x b E d \ Λ,y b E d x b y b =, z O d Λ with x b z = y b z = 1}. 14) With these estimates and knowing that BΛ Λ C N, the proof from Step in [14] follows now immediately. Step 3 The proof for Step 3 follows the same way as in [14]. The desired estimate follows now from the fact that T d α=1 [ E P ηt E e E α) η ) ] T t EeE α) dt gt) dt. 15) Theorem 1 For every u R d, there exists at most one µ E ext S E) ũ. Proof. By using Lemma, the proof follows the same arguments as in [14], so it will be omitted. 4. Proof of Theorem 17 Proof. Note first that any µ GH) is reversible under the dynamics η t defined by the 13). In particular, G S. Suppose now that there exist µ, µ GH) ergodic such that E µ η t e α )) = u α,α = 1,... d for u R d. Note first that E µ η E t e E α )) = E µ η E t e E α )) = ũ α,α = 1,... d. Hence, from Lemma 4, we get that µ E d, µ E d G E H ) ) such that for all finite Λ Z d, we have H ) L. EΛ d Since for all η E χ E, with η E b) = φy b ) φx b ), b B E d, we can write η E b) = ηb 1 ) + ηb ), b 1,b B Z d, ergodicity and invariance under the even shifts for µ E d, µ E d follow immediately from the similar properties for µ, µ. We also have reversibility for the even see for example [15]). Therefore µ E d, µ E d ext S E) ũ, so we can apply Theorem 1 to get µ E d = µ Ed. Then for any A F, we have E BZ d µ1 A F ),E µ1 BE d A F ) F BE d B E. From Lemma 7, we have d E µ 1 A F BE d ) = E µ1 A F BE d ) 16) 1

22 and thus µa) = E µ 1 A ) = E µ E µ 1 A F BE d )) = E µe µ 1 A F BE d )) = E µ E µ 1 A F BE d )) = E µa) = µa). 17) Remark Note that as an immediate consequence of the results in Sections 3., 3.3 and of Remarks 15 and 16, we can also adapt the reasoning above to get the uniqueness of the ergodic component in the case where, for all i I, U i are strictly convex and non-symmetric. 5 Covariance We will extend in this section the covariance estimates of [9] to the class of non-convex potentials U i = V i + g i which satisfy 15) such V i and g i satisfy 16), 17) and 64). Let µ u P χ) be the unique shift invariant measure, ergodic with respect to shifts such that E µu ηe α )) = u α,α = 1,... d. Let { B O Z := b = x d b,y b ),b B Z d such that x b O d}. 18) Let F C 1 b RBO Z d ). We will denote for every i I and for every b = x,x + e i ) B O Z d by b F = x,x+ei )F = ei φx) F φ) and x,x+e i F = sup x,x+ei F φ). 19) φ Theorem 3 Let F,G C 1 b RBO Z d ). Then there exists C > such that Proof. We have where cov µu F φ),g φ)) C b,b B O Z d cov µu F φ),g φ)) = E µu [ cov µu F φ),g φ) F BE d ) ] E µu F φ) F BE d ) = b F b G 1 + b 1 b 1 d. 13) + cov µu E µu [F φ) F BE d ],E µ u [G φ) F BE d ] ), 131) F φ) µ u ) y φy + e k ) φy) F BE )dφy), 13) d y O d since the conditional measure is a product measure; a similar formula holds for G. Note that under µ F BE d ), the gradients φ ix),x O d,i I) are independent. Thus cov µu F φ),g φ) F BE d ) b F b G var µu φb) F ) BE d b B O Z d σ b F b G, 133) b B O Z d

23 where for the first inequality we used a result in [1] and for the last inequality we used 85). Next, in view of [9] cov µu ˆF,Ĝ) c b,b B E d b ˆF b Ĝ 1 + b 1 b 1 d, 134) where ˆF = E µu [F φ) F ] and Ĝ = E BE d µ u [G φ) F BE ]. 135) d We need to estimate now b ˆF and b Ĝ. Suppose that b = x + e k,x + e j ) for some x O d and j,k I. But b ˆF = b E µu [F φ) F ] = BE d b F φ) µ u ) y φy + e k ) φy) F )dφy) BE d y O d = b F φ) 1 Z θy)) exp ) U i φy) φy + e k ) φy + e i ))) dφy) y O d = b F φ) 1 Z θy)) exp ) U i φy) φy + e k ) φy + e i ))) dφy) y O d cov µu F φ), b U i φy) φy + e k ) φy + e i ))) F B E. 136) d y O d To estimate the above equation, we will use now the following simple change of variables φx + e k ) + φx + e j ) = a and φx + e k ) φx + e j ) = b, 137) therefore φx + e k ) = a+b and φx + e j ) = a b. Then b F φ) = s {k,l} {z E d : z x+e s) =} F φ) z,x+s) φ φz) φx + e s )), 138) b where we denoted by F φ) the partial derivative D l F such that l is the indice which gives the z,x+es) position in F of φz) φx + e s ). Since a similar formula holds for the term differentiated in the covariance, from 136) we obtain b ˆF 1 s {k,l} {z E d : z x+e s) =} + cov µu F φ), F φ) z,x+es) φ U i φy) φy + e k) φy + e i ))) F B E d {y O d : y+e i b or y+e k b} By using 133) applied to F φ) and to {y O d : y+e i b or y+e k b} 139) U i φy) φy + e k) φy + e i ))), by using the fact that U i C + C and a similar formula to to the one used to derive 138), we 3

24 get cov µu 8d C +C ) F φ), U i φy) φy + e k) φy + e i ))) F B E d b F var µu φb) F ) BE d 8d σ C +C ) {y O d : y+e i b or y+e k b} b F. b B O Z d y+e k b or y+e i b We also used in 14) the fact that b {y O d : y+e i b or y+e k b} if neither y + e k, nor y + e i are vertices of the bond b. Note now that δ x F) x,x+ei )F and δ x G) b B O Z d y+e k b or y+e i b 14) ) U i φy) φy + e k) φy + e i ))) = x,x+ei )G, with δ x F) = sup φ φx) Fφ). 141) The statement of the theorem follows now from 139), 14), 141), 133) and 134). 6 Scaling Limit We will extend next the scaling limit results results from [16] to a class of non-convex potentials. Theorem 4 Let u R d. Let µ u GH) ergodic with E µu [ηe i )] = u i Assume U i = V i + g i, where U i satisfy 15) and V i and g i satisfy 16), 17) and 64). Set S ɛ f) = ɛ d/ fxɛ)φx + e i ) φx) u i ), 14) x Z d where f C Rd ). Then where σ uf) >. S ɛ f) N,σu f)) as ɛ, 143) Proof. Let us write E d for the even sites and O d for the odd sites of Z d. Then S ɛ f) = ɛ d/ x Z d fxɛ)φx + e i ) φx) u i ) = ɛ d/ x E d fxɛ)φx + e i ) φx) u i ) ɛ d/ x E d fxɛ)φx + e i ) φx + e i ) u i ) + ɛ d/ = ɛ d/ x E d fxɛ)φx + e i ) φx) u i ) x O d fxɛ)φx + e i ) φx) u i ) +ɛ ) fx d/ + e i )ɛ) fxɛ) φx + e i ) φx + e i ) u i ) = Sɛ e f) + R ɛf). 144) x E d 4

25 Now we can show CLT for Sɛ e f) since the summation is concentrated on the even sites; the proof uses the same arguments as in [16] and is based on the RW representation. Also, since by Theorem 3 cov µu i φx), j φy)) C x y + 1) d, 145) we have var µu R ɛ f)) ɛ d i fxɛ) i fyɛ) cov µu φx + e i ) φx),φy + e i ) φy)) x,y E d ɛ d C i fxɛ) i fyɛ) x y + 1) d, 146) x,y E d where i fxɛ) = fx + e i )ɛ) fxɛ). Expanding fx + e i )ɛ) in Taylor expansion around xɛ, we have i fxɛ) = D i fa)ɛ, 147) for some a R d. As f C Rd ), there exist M,N > such that for all x R d with ɛx N we have fɛx) M, D i fɛx) M and both functions equal to for ɛx > N. Therefore var µu R ɛ f)) ɛ d+ M C It now remains to evaluate the sum x,y E d, ɛx N, ɛx N x,y E d, ɛx N, ɛx N 1 x y + 1) d y E d, ɛy N x,y E d, ɛx N, ɛx N 1 x y +1) d. But N+1 ɛ N+1 ɛ N+1 ɛ... N+1 ɛ 1 x y + 1) d. 148) dx 1 dx... dx d d i=1 x i y i + 1) d ɛ Cd,N)log 1 + dn/ɛ) dncd,n)ɛ, 149) where Cd,N) is a positive constant depending on d and N. It follows that R ɛ f) as ɛ in probability. 7 Surface tension 7.1 Surface tension on the torus We will extend here to the family of non-convex potentials satisfying 15), 16), 17) and 64), the surface tension strict convexity result from [14] and [1]. Additionally, in Theorem 8 we prove a series of surface tension equalities, which are important for the derivation of the hydrodynamic limit. To study the convexity properties of the surface tension as a function of the tilt u) for nonconvex gradient models on a lattice, we will work on the torus, instead of the finite box on the lattice Z d. Thus, let T d N = Z/NZ)d = Z d mod N) be the lattice torus in Z d and let u R d. Then, we define the surface tension on the torus T d N as σ β T N u) = 1 β ZT T d log N u) N d Z β T N ), 15) 5

26 where Z β T N u) = R Td N exp βh TN φ +,u )) dφx) 151) x T d N \{} and where H TN is the Hamiltonian on the torus T d N given by H TN φ) = U i i φx)) = [V i i φx)) + g i i φx))]. 15) x T d N x+e i T d N x T d N x+e i T d N Note that we define u i = u i for i = 1,,...,d. Just as before, let us label the vertices of the torus as odd and even; let the set of odd vertices be O d N and the set of even vertices be Ed N. Then we can of course first integrate all the odd coordinate first and then: Z β T N u) = exp βh TN φ +,u ) dφx) dφx) R Ed N R Od N x O d N x E d N \{} = exp βh E R Ed d φ,u)) dφx), 153) N N x E d N \{} where H EN φ,u) is the induced Hamiltonian on the even. It is easy to see that Then, defining the even surface tension on E d N as where Z β E N u) = H EN φ,u) = H EN φ +,u,). 154) R Ed N σ β E N u) = 1 E d N log Z β E N u) Z β E N ), 155) exp βh EN φ +,u,)) we obtain the following result by integrating out the odds Lemma 5 x E d N \{} dφx), 156) σ β E N u) = 1 σβ T N u). 157) We will next prove strict convexity for the even surface tension, uniformly in N. Lemma 6 Suppose that V i,g i C R) such that they satisfy 16), 17) and 64). Then, for all N = k, we have D σ β E N u) C E d N Id, u R d. 158) That is, the even surface tension is uniformly convex, uniformly in N. Proof. First note that if N = k, we can write H EN φ,u) as H EN φ,u) = F x θ,u), 159) x O d N where for all x O d N F x θx),u) = log e β U i i φx)+u i ) dφx) 16) R 6

27 and where, just as in 46) Note that for all i I, we have u i = u i. Then σ β E N u) = 1 E d N log θx) = φx + e 1 ),...,φx e d )). 161) E e d x O d N N E e N d x O d N F xθx),u) x+e i E d N F xθx),) x+e i E d N dφx + e i ). 16) dφx + e i ) As the denominator of σ EN u) doesnt depend on u, it is enough to focuss on the term R EN u) := log e x O d F xθx),u) N dφx + e i ). 163) E d N x+e i E d N Note now that by Theorem 1, we have that for each x O d Λ, F x is convex, that is D F x θ) θ) ) θ) c 1 i,j I, i j θx + ei ) θx + e j ). 164) Because by Theorem 1 the F x fullfil the random walk representation, we can apply to R N Lemma 3. in [8], 164) and the fact that for all i I, we have u i = u i, to get the statement of the lemma. We consider the finite volume Gibbs measures µ N,u Pχ T d N ) with periodic boundary conditions which, for each u R d, are defined by µ N,u d η) = Z 1 N,u exp 1 V ηb) + u b ) d η N Pχ BZ N). 165) d b B Z d N Here d η N is the uniform measure on the affine space χ T d and Z N,u is the normalizing constant. N The law of {ηb) := ηb) + u b } under µ N,u is denoted by µ N,u. Lemma 7 µ N,u converges weakly to µ u ext G. Proof. Tightness of the family {µ N,u } N is known for non-convex potentials with quadratic growth at see Remark 4.4 page 15 in [15]). Therefore a limiting measure exists by taking N along a suitable subsequence. From now on and using Theorem 17, the proof follows the same reasoning as the proof of Theorem 3. in [14]. In particular, because of the uniqueness of ergodic gradient Gibbs measures for each u, µ N,u converges weakly to µ u. Let σ T d N = D 1 σ T dn,...,d d σ T dn ), where D i σ T d N = σ T d N u i,i = 1,... d. 166) Theorem 8 Suppose that V i,g i C R) such that they satisfy 16), 17) and 64) and such that for all i I, U i are symmetric. Then we have a) lim N σβ T N u) = σ T u), σ T C 1 R d ); 167) 7

28 b) σ T is strictly convex as a function of u; c) E µu [ηb)] = u b ; d) E µu [U i ηe i)] = D i σ T u), for all i = 1,... d; e) E µu [ d i=1 ηe i)u i ηe i)] = u σ T u) + 1, for all i = 1,... d; f) σu) σv) C u v for some C >. Proof. a) Noting from Lemma 7 that µ N,u converges weakly to µ u as N and using the tightness of the family {µ N,u } N, the proof now follows the same steps as the proof of Theorem 3.4.) in [14]. b) Since by a) lim N σβ T N u) = σ T u), 168) every subsequence of σ β T N u) will converge to σ T u), in particular for N = k. The statement of the theorem follows ) immediately by using now Lemma 5 and Lemma 6 applied to the subsequence σ β T N u), with N = k. N c), d) and e) follow just as in [14], so their proofs will be omitted. f) Let N = k. Define µ E N,u dφe ) = 1 Z E N,u exp F x θx),u) dφ E N, 169) x O N where dφ En = x E d N \{} φx), ZE N,u is the normalizing constant and F x and θx) are defined as in 16) and 46), respectively. Due to the fact that the random walk representation holds on the set of the evens and to Theorem 1, one can show as in [14] that for N = k, µ E N,u converges weakly to µ E u ext S E), where the same notations as in the uniqueness of ergodic ũ component section apply. Note now from 16) that E µn,u [ U i i φx) ] = E µ E N,u [ D i F x θx),u) ], 17) where x O d N. Using now d), the weak convergence of µ N,u to µ u and the weak convergence of µ E N,u to µe u, we get [ E µeu D i F x θx),u) ] = D i σ T u). 171) Uisng the random walk representation and Taylor expansion, we have D i F x θx)) D i F x θx)) c + φx + e k ) φx + e k ). 17) The bound in f) is now a simple consequence of 171), 17) and Lemma. k I 8

29 7. Equality between the surface tension on the torus and the surface tension on the box Let Λ N = [ N,N] d Z d be a cube of side length N + 1 in Z d with boundary Λ N = {x = x 1,...,x n ) x α = N for at least one α,α = 1,... d}. We enforce a tilt u = u 1,...u d ) R d by setting φx) = x u for x Λ N. Then, we define the surface tension on Λ N as Z β Λ N u) = σ β Λ N u) = 1 Λ N d log Z β Λ N u) Z β, where 173) Λ N ) R Λd N exp βh ΛN φ)) x Λ N dφx) y Λ N δφy) u y), 174) and where H ΛN is the Hamiltonian given by 14). Noting that C 1 C U i C, the proof for the existence of the surface tension on the box follows the same steps as in [14]. We thus have Theorem 9 exists and is independent of the chosen sequence Λ. lim σ Λ u) = σ B u) 175) Λ Z d To prove the equality between the surface tension on the torus and the surface tension on the box, we follow the same steps as Lemma II. in [14]. Note that we will also use the same reasoning as from our covariance section in order to estimate the variance needed for the proof. Lemma 3 Let U i, i I, be symmetric potentials. Then we have where σ T u) is the one from Theorem 8. 8 Appendix σ B u) = σ T u), 176) 8.1 Counter-Example to Random Walk Representation We will next outline a method to check if the RW representation holds for large β; we will also provide a counter-example when the RW doesnt hold for very large β. Let µdx) = 1 β exp βhs))ds. We assume that there exists a unique minimum s such that H s) = min s Hs), where H s) >. 177) Let U 1,U C R), with U i s), for i = 1,. Note that cov µβ U 1,U ) = 1 Fs 1,s )µ β ds 1 )µ β ds ), 178) with Fs 1,s ) = [U 1 s 1 ) U 1 s )] [U s 1 ) U s )]. Note now that Fs 1,s ) = F s, s) + s1 F s, s)s 1 s) + s F s, s)s s) + 1 s1 F s, s)s 1 s) + s1 s F s, s)s 1 s)s s) + s F s, s)s s) ) + o1), 179) 9