Decay of covariances, uniqueness of ergodic component and scaling limit for a class of <span class="mathjax-formula">$\nabla \phi $</span> systems with non-convex potential

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Decay of covariances, uniqueness of ergodic component and scaling limit for a class of ∇ φ systems

with non-convex potential ¹

Codina Cotar

and Jean-Dominique Deuschel

aTU München, Zentrum Mathematik, Lehrstuhl für Mathematische Statistik, Boltzmannstr. 3, 85747 Garching, Germany.

E-mail:cotar@ma.tum.de

bTU Berlin, Fakultät II, Institut für Mathematik, Strasse des 17. Juni 136, D-10623 Berlin, Germany. E-mail:deuschel@math.tu-berlin.de Received 27 April 2010; revised 11 March 2011; accepted 23 May 2011

Abstract. We consider a gradient interface model on the lattice with interaction potential which is a non-convex perturbation of a convex potential. Using a technique which decouples the neighboring vertices 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.

Résumé. Nous considérons un modèle d’interfaces de type gradient indexé par le réseau avec une interaction donnée par la pertubation non convexe d’un potentiel convexe. En utilisant une technique qui découple les sites pairs et impairs, nous démontrons pour une classe de potentiels non convexes l’unicité de la composante ergodique, de la mesure de Gibbs du gradient, la décroissance des covariances, la loi limite centrale et la stricte convexité de la tension superficielle.

MSC:60K35; 82B24; 35J15

Keywords:Effective non-convex gradient interface models; Uniqueness of ergodic component; Decay of covariances; Scaling limit; Surface tension

1. Introduction 1.1. The setup

Phase separation inR^d+1can be described by effective interface models, where interfaces are sharp boundaries which separate the different regions of space occupied by different phases. In this class of models, the interface is modeled as the graph of a random function fromZ^dtoZorR(discrete or continuous effective interface models). For more on interface models, see the reviews by Funaki [20] or Velenik [27]. In this setting we ignore overhangs and forx∈Z^d, we denote byφ(x)∈Rthe height of the interface above or below the sitex. LetΛbe a finite set inZ^dwith boundary

∂Λ:=

x /∈Λ,x−y =1 for somey∈Λ

, wherex−y = d i=1

|xi−yi|forx, y∈Z^d (1) and with given boundary conditionψsuch thatφ(x)=ψ (x)forx∈∂Λ; a special case of boundary conditions are thetiltedboundary conditions, withψ (x)=x·ufor allx∈∂Λ, and whereu∈R^dis fixed. LetΛ:=Λ∪∂Λand let

1Supported by the DFG-Forschergruppe 718 ‘Analysis and stochastics in complex physical systems.’

dφΛ=

x∈Λdφ(x)be the Lebesgue measure overR^Λ. For a finite regionΛ⊂Z^d,the finite volume Gibbs measure ν_Λ,ψ onR^Zd with boundary conditionψfor the field ofheight variables(φ(x))_x∈Zd overΛis defined by

νΛ,ψ(dφ)= 1 Z_Λ,ψ exp

−βHΛ,ψ(φ)

dφΛδψ(dφ_Zd\Λ) (2)

with

ZΛ,ψ=

R^Zd exp

−βHΛ,ψ(φ)

dφΛδψ(dφ_Zd\Λ), and whereδ_ψ(dφ_Zd\Λ)=

x∈Z^d\Λδ_ψ(x)(dφ(x))and determines the boundary condition. Thus,ν_Λ,ψis characterized by the inverse temperatureβ >0 and the HamiltonianHΛ,ψ onΛ, which we assume to be of gradient type:

H_Λ,ψ(φ)=

i∈I

x,x+e_i∈Λ

U

∇iφ(x) +2

i∈I

x∈Λ,x+e_i∈∂Λ

U

∇iφ(x)

, (3)

where the sum insideΛis over ordered nearest neighbours pairs(x, x+e_i). We denoted by I= {−d,−d+1, . . . , d} \ {0}

and we introduced for eachx∈Z^dand eachi∈I, the discrete gradient

∇iφ(x)=φ(x+e_i)−φ(x),

that is, the interaction depends only on the differences of neighboring heights. Note thate_i, i=1,2, . . . , d, denote the unit vectors ande_−i= −e_i. A model with such a Hamiltonian as defined in (3), is called a massless model with a continuous symmetry (see [20]). The potentialU∈C²(R)is a symmetric function with quadratic growth at infinity:

U (η)≥A|η|²−B, η∈R, (A0)

for someA >0, B∈R.

1.2. General definitions and notation 1.2.1. φ-Gibbs measures

ForA⊂Z^d, we shall denote byFAtheσ-field generated by{φ(x): x∈A}.

Definition 1.1 (φ-Gibbs measure onZ^d). The probability measureν∈P (R^Zd)is called a Gibbs measure for the φ-field with given HamiltonianH:=(H_Λ,ψ)

Λ⊂Z^d,ψ∈R^Zd (φ-Gibbs measure for short),if its conditional probability ofFΛ^c satisfies the DLR equation

ν(·|FΛ^c)(ψ )=ν_Λ,ψ(·), ν-a.e.ψ, for every finiteΛ⊂Z^d.

It is known that theφ-Gibbs measures exist under condition (A0) when the dimensiond≥3, but not ford=1,2, where the field “delocalizes” asΛZ^d (see [15]). An infinite volume limit (thermodynamic limit) forν_Λ,ψ when ΛZ^dexists only whend≥3.

Notation for the bond variables onZ . Let Z^d_∗

:=

b=(x_b, y_b)|x_b, y_b∈Z^d,x_b−y_b =1, bdirected fromx_btoy_b

; note that each undirected bond appears twice in(Z^d)^∗. Let

Λ^∗:=

Z^d_∗

∩(Λ×Λ), ∂Λ^∗:=

b=(x_b, y_b)|x_b∈Z^d\Λ, y_b∈Λ,x_b−y_b =1 and

Λ^∗:=

b=(x_b, y_b)∈ Z^d_∗

|x_b∈Λory_b∈Λ .

Forφ=(φ(x))_x∈Zd andb=(x_b, y_b)∈(Z^d)^∗, we define the height differences∇φ(b):=φ(y_b)−φ(x_b). The height variablesφ= {φ(x);x∈Z^d}onZ^d automatically determines a field of height differences∇φ= {∇φ(b);b∈ (Z^d)^∗}. One can therefore consider the distributionμof∇φ-field under theφ-Gibbs measureν. We shall callμthe

∇φ-Gibbs measure. In fact, it is possible to define the∇φ-Gibbs measures directly by means of the DLR equations and, in this sense,∇φ-Gibbs measures exist for all dimensionsd≥1.

A sequence of bondsC= {b⁽¹⁾, b⁽²⁾, . . . , b⁽ⁿ⁾}is called achainconnectingx andy,x, y∈Z^d, ifx_b1=x, y_b(i)= x_b(i+1) for 1≤i≤n−1 andy_b(n)=y. The chain is called aclosed loopify_b(n)=x_b(1). Aplaquetteis a closed loop A= {b⁽¹⁾, b⁽²⁾, b⁽³⁾, b⁽⁴⁾}such that{x_b(i), i=1, . . . ,4}consists of 4 different points.

The fieldη= {η(b)} ∈R^(Zd)∗, b∈(Z^d)^∗,is said to satisfythe plaquette conditionsif

η(b)= −η(−b) for allb∈ Z^d_∗

and

b∈A

η(b)=0 for all plaquettesAinZ^d, (4) where−bdenotes the reversed bond ofb. Let

χ=

η∈R^(Zd)∗which satisfy the plaquette condition

(5) and letL²_r, r >0, be the set of allη∈R^(Zd)∗such that

|η|²r:=

b∈(Z^d)^∗

η(b) ²e^−2rxb<∞.

We denoteχ_r=χ∩L²_r equipped with the norm| · |r. Forφ=(φ(x))_x∈Zd andb∈(Z^d)^∗, we defineη^φ(b):= ∇φ(b).

Then∇φ= {∇φ(b)}satisfies the plaquette condition. Conversely, the heightsφ^{η,φ (0)}∈R^Zd can be constructed from height differencesηand the height variableφ(0)atx=0 as

φ^{η,φ (0)}(x):=

b∈C0,x

η(b)+φ(0), (6)

whereC0,x is an arbitrary chain connecting 0 andx. Note thatφ^{η,φ (0)}is well-defined ifη= {η(b)} ∈χ.

Definition of∇φ-Gibbs measures. We next define the finite volume∇φ-Gibbs measures. For everyξ∈χand finite Λ⊂Z^dthe space of all possible configurations of height differences onΛ^∗for given boundary conditionξis defined as

χ_Λ∗,ξ= η=

η(b)

b∈Λ^∗;η∨ξ∈χ ,

whereη∨ξ∈χis determined by(η∨ξ )(b)=η(b)forb∈Λ^∗and=ξ(b)forb /∈Λ^∗.

Remark 1.2. Note that whenZ^d\Λis connected,χ_Λ∗,ξ is an affine space such thatdimχ_Λ∗,ξ= |Λ|.Indeed,fixing a pointx₀∈/Λ,we consider the mapχ_Λ∗,ξ→R^Λ,such thatη→φ= {φ(x)} ∈R^Λ,withφ(x)defined by

φ(x)=

b∈C_x0,x

(η∨ξ )(b)

for a chainCx₀,x connectingx0andx∈Λ.This map then well-defined and an invertible linear transformation.

Definition 1.3 (Finite volume∇φ-Gibbs measure). The finite volume∇φ-Gibbs measure inΛ(or more precisely,in Λ^∗)with given HamiltonianH:=(H_Λ,ξ)_Λ⊂Zd,ξ∈χand with boundary conditionξ is defined by

μ_Λ,ξ(dη)= 1 Z_Λ,ξ exp

−β

b∈Λ^∗

U η(b)

dηΛ,ξ∈P (χ_Λ∗,ξ),

wheredηΛ,ξ denotes the Lebesgue measure on the affine spaceχ_Λ∗,ξandZ_Λ,ξ is the normalization constant.

LetP (χ )be the set of all probability measures onχand letP2(χ )be thoseμ∈P (χ )satisfyingE^μ[|η(b)|²]<∞ for eachb∈(Z^d)^∗.

Remark 1.4. For everyξ ∈χ and a∈R,let ψ =φ^ξ,a be defined by (6) and consider the measureν_Λ,ψ.Then μΛ,ξ is the image measure ofνΛ,ψ under the map{φ(x)}x∈Λ→ {η(b):= ∇(φ∨ψ )(b)}_b∈Λ∗and where we defined (φ∨ψ )(x):=φ(x)forx∈Λand(φ∨ψ )(x):=ψ (x)forx /∈Λ.Note that the image measure is determined only by ξ and is independent of the choice ofa.LetK_Λ^ψ:{φ(x)}x∈Z^d → {η(b)}b∈(Z^d)^∗,withη(b):= ∇(φ∨ψ )(b).

Definition 1.5 (∇φ-Gibbs measure on(Z^d)^∗). The probability measureμ∈P (χ )is called a Gibbs measure for the height differences with given HamiltonianH:=(H_Λ,ξ)_Λ⊂Zd,ξ∈χ(∇φ-Gibbs measure for short),if it satisfies the DLR equation

μ(·|F_(Z^d₎∗\Λ^∗)(ξ )=μΛ,ξ(·), μ-a.e.ξ, (7)

for every finiteΛ⊂Z^d,whereF_(Z^d₎∗\Λ^∗stands for theσ-field ofχgenerated by{η(b), b∈(Z^d)^∗\Λ^∗}.

Remark 1.6. Proving the DLR equation(7)is equivalent to proving that for every finiteΛ⊂Z^dand for allF∈Cb(χ ) we have

χ

μ(dξ )

χ_Λ∗,ξ

μΛ,ξ(dη)F (η)=

χ

μ(dη)F (η). (8)

(For a proof of this equivalence,see Remark1.24from[21].) With the notations from (3) and Definition1.3, let

Gβ(H ):=

μ∈P2(χ ): μis∇φ-Gibbs measure on Z^d_∗

with given HamiltonianH .

Remark 1.7. Throughout the rest of the paper,we will use the notationφ, ψ to denote height variables andη, ξ to denote height differences.

Shift-invariance and ergodicity. For x ∈Z^d, we define the shift operators: σ_x:R^Zd →R^Zd for the heights by σ_xφ(y)=φ(y−x)for y∈Z^d andφ∈R^Zd, andσ_x:R^(Zd)∗→R^(Zd)∗ for the bonds by(σ_xη)(b)=η(b−x), for b∈(Z^d)^∗ andη∈χ. Thenshift-invariance andergodicityfor μ (with respect to σ_x for allx ∈Z^d) is defined in the usual way (see for example p. 122 in [20]). We say that the shift-invariantμ∈P2(χ ) has a giventiltu∈R^d if Eμ(η(b))= u, yb−xbfor all bondsb=(xb, yb)∈(Z^d)^∗.

Our state spaceR^Zd being unbounded, gradient interface models experience delocalization in lower dimensionsd= 1,2, and no infinite volume Gibbs state exists in these dimensions (see [15]). Instead of looking at the Gibbs measures of the(φ(x))_x∈Zd, Funaki and Spohn proposed to consider the distribution of the gradients(∇iφ(x))_i∈I,x∈Zd under ν (see Definition1.5) in thegradient Gibbs measuresμ, which in view of the Hamiltonian (3), can also be given in terms of a Dobrushin–Landford–Ruelle (DLR) description. Note that infinite volume gradient Gibbs measures exist in all dimensions, in particular for dimensions 1 and 2, which is one of the reasons that Funaki and Spohn introduced them. For a good background source on these models, see Funaki [20].

Assuming strict convexity ofU:

0< C₁≤U≤C₂<∞, (9)

Funaki and Spohn showed in [19] the existence and uniqueness of ergodic gradient Gibbs measures for every fixed tiltu∈R^d, that is, ifEμ(∇iφ(x))=u_i for all nearest-neighbour pairs(x, x+e_i)(see also [10,26]). Moreover, they also proved that the corresponding free energy, or surface tension,σ (u)∈C¹(R^d)is convex inu; the surface tension, defined in Section 7 of our paper, physically describes the macroscopic energy of a surface with tiltu, i.e., ad- dimensional hyperplane located inR^d+1with normal vector(−u,1)∈R^d+1. Both these results (ergodic component and convexity of surface tension) were used in [19] for the derivation of the hydrodynamical limit of the Ginzburg–

Landau model.

In fact under the strict convexity assumption (9) ofU, much more 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 quadraticU. In particular, Brydges and Yau [7] (in the case of small analytic perturbations of quadratic potentials), Naddaf and Spencer [25] (in the case of strictly convex potentials and tiltu=0) and Giacomin, Olla and Spohn [22] (in the case of strictly convex potentials and arbitrary tilt u) showed that the rescaled gradient field converges weakly asε0 to a continuous homogeneous Gaussian field, that is

S_ε(f )=ε^d/2

x∈Z^d

i∈I

∇iφ(x)−u_i

f_i(εx)→N

0, Σ_u²(f )

asε→0, f∈C₀^∞ R^d;R^d

, (10)

where the convergence takes place under ergodicμwith tiltu(see Theorem 2.1 in Giacomin, Olla and Spohn [22] for an explicit expression ofΣ_u²(f )in (10) in the case with arbitrary tilt and see Biskup and Spohn [3] for similar results in the non-convex case). This central limit theorem derived at standard scalingε^d/2, is far from trivial since as shown in Delmotte and Deuschel [11], the gradient field has slowly decaying, non-absolutely summable covariances of the algebraic order

covν

∇iφ(x),∇jφ(y) ∼ C

1+ x−y^d. (11)

All the above-mentioned results are proved under the essential assumption of strict convexity of the potentialU, which assumption is necessary for the application of the Brascamp–Lieb inequality and of the Helffer–Sjostrand random walk representation (see [20] for a detailed review of these methods and results). While strict convexity is crucial for the proofs, one would expect some of these results to be valid under more general circumstances, in particular also for some classes of non-convex potentials. However, so far there have been very few results on non-convex potentials.

This is where the focus of this paper comes in, which is to extend the results known for strictly convex potentials to some classes of non-convex potentials.

We will briefly summarize next the state of affairs regarding results for non-convex potentials, in the different regimes at inverse temperatureβ. At low temperature (i.e. largeβ) using the renormalization group techniques devel- oped by Brydges [6], Adams et al. [1] show in on-going work for a class of non-convex potentials, the strict convexity of the surface tension for small tiltu. At moderate temperature (β=1), Biskup and Kotecký [2] give an example of a non-convex potentialU for which uniqueness of the ergodic gradient Gibbs measuresμfails. The potentialUcan be described as the mixture of two Gaussians with two different variances. For this particular case ofU, [2] prove co-existence of two ergodic gradient Gibbs measures at tiltu=0 (see also Fig.4and example (a) in Section3.2). See

also the work of Fröhlich and Spencer [17,18] in relation to the Coulomb gas, and the theory based on the infrared- bound (e.g. Fröhlich, Simon and Spencer [16]). For high temperature (i.e. smallβ), we have proved in a previous paper with S. Mueller [8] strict convexity of the surface tension in a regime similar to (A2) below. Our potentials are of the form

U

∇iφ(x)

=V

∇iφ(x) +g

∇iφ(x) , whereV , g∈C²(R)are such that

C1≤V≤C2, 0< C1< C2 and −C0≤g≤0 withC0> C2. Specifically, we assumed in [8] that

4

π(12dC)¯ ^1/2 βC₁ 1

C1

g

L¹(R)≤ 1

2, whereC¯=max C0

C1

,C2

C1−1,1

.

The method used in [8], based on two scale decomposition of the free field, gives less sharp estimates for the tem- perature than our current paper as the estimates also depend onC₀. However, at this point it is not clear whether the method introduced in [8] could yield any other result of interest than the strict convexity of the surface tension.

The aim of our current paper is to use an alternative technique from the one we used in [8] and relax the strict convexity assumption (9) to obtain much more than just strict convexity of the surface tension; more precisely, we also prove uniqueness of the ergodic component at every tiltu∈R^d, central limit theorem of form as given in (10) and decay of covariances as in (11). As stated above, the hydrodynamical limit for the corresponding Ginzburg–Landau model should then essentially follow from our results. Our main results are proven under the assumption that

C₁≤V≤C₂, 0< C₁< C₂ and −∞< g≤0 (A1)

and that the inverse temperatureβis sufficiently small, that is if β^1/(2q)g

L^q(R)< (C1)^3/2

2C₂^(q+1)/(2q)(2d)^1/(2q) for someq≥1, (A2)

or if

β^3/4g

L²(R)≤ (C₁)^3/2

2(C2)^5/4(2d)^3/4. (A3)

The condition (A1) withg≤0 may look a bit artificial, but as we elaborate in Remark3.12in Section3below, any perturbationg∈C²with compact support can be substituted for theg≤0 assumption in (A1). Note that in contrast to the condition in our previous paper [8],gL^∞(R)can be arbitrarily large as long asgL^q(R)is small. Note also that using an obvious rescaling argument (see Remark3.8), we can always reduce our assumption (A1) to the case β=C1=1; then (A2), respectively (A3), states that our condition is satisfied whenever the perturbationgis small in theL^q(R), respectivelygis small in theL²(R)sense.

Our main result is the following

Theorem 1.8 (Uniqueness of an ergodicμ_u). LetU=V +g,whereUsatisfy(A0)andV andgsatisfy(A1)and (A2)or(A1)and (A3).Then for everyu∈R^d,there exists at most one ergodic,shift-invariantμ_u∈Gβ(H )with a given tiltu∈R^d.

LetF ∈C¹_b(χ_r), whereC_b¹(χ_r)denotes the set of differentiable functions depending on finitely many coordinates with bounded derivatives and whereχ_r was defined in Section1.2.2. Forη, η∈χ, let

εlim→0

F (η+εη)−F (η)

ε =

DF (η), η

=

b∈(Z^d)^∗

α(b)η(b).

∂bF (η):=α(b) and ∂bF∞=sup

η∈χ

∂bF (η) . (12)

Another result we prove for our class of non-convex potentials is

Theorem 1.9 (Decay of covariances). Letu∈R^d.AssumeU=V +g,whereUsatisfies(A0)andV andgsatisfy (A1)and(A2)or(A1)and(A3).LetF, G∈C_b¹(χr).Then there existsC >0such that

covμ_u

F (η), G(η) ≤C

b,b∈(Z^d)^∗

∂_bF_∞∂_bG_∞

1+ x_b−x_b^d , (13)

whereb=(x_b, y_b)andb=(x_b, y_b).

We also prove

Theorem 1.10 (Central limit theorem). Letu∈R^d.AssumeU=V+g,whereUsatisfies(A0)andV andgsatisfy (A1)and(A2)or(A1)and(A3).Set

S_ε(f )=ε^d/2

x∈Z^d

i∈I

∇iφ(x)−u_i f_i(εx),

wheref ∈C₀^∞(R^d;R^d).Then S_ε(f )⇒N

0, Σ_u²(f )

asε→0,

whereΣ_u²(f )can be identified explicitly as in Theorem2.1in[22],Σ_u²(f )=0forf=0,and⇒signifies convergence in distribution.

Moreover, we extend in Theorem7.3the results of strict convexity of the surface tension from [14] and [19] to the family of non-convex potentials satisfying (A0), (A1) and (A2).

Even though our results are obtained for the high temperature case, previously only our results in [8] were known for the non-convex case. Also, the proofs of this paper require some crucial observations not made before. Moreover, in our main result Theorem1.8, we prove uniqueness of ergodic gradient Gibbs measuresμwith a given arbitrary tilt u∈R^dfor the class of non-convex potentials satisfying (A0), (A1) and (A2). To the best of our knowledge, this is the first result where uniqueness of ergodic gradient Gibbs measuresμis proved for a class of non-convex potentialsU.

For potentials that are mixtures of Gaussians as considered in Biskup and Kotecký [2], they prove non-uniqueness of ergodic gradient Gibbs measures for tiltu=0 in theβ=1 regime. For the same example, we prove uniqueness of ergodic gradient Gibbs measures for given arbitrary tiltuin the high temperature regime. Therefore, our result also highlights the existence of phase transition for these models in different temperature regimes.

The basic idea relies on a one-step coarse graining procedure, in which we consider the marginal distribution of the gradient field restricted to the even sites, which is also a gradient Gibbs field. The corresponding Hamiltonian, although no longer a two-body Hamiltonian, is then obtained via integrating out the field at the odds sites. We can integrate out the fieldφat all odd sites, using the fact that they are independent conditional on the fieldφat even sites, which is a consequence of the bi-partiteness of the graphZ^dwith nearest-neighbor bonds. The crucial step, which is similar to the idea of our previous paper [8], is that strict convexity can be gained via integration at sufficiently high temperature (see also Brascamp et al. [5] for previous use of the even/odd representation). The essential observation is that we can formulate a condition for this multi-body potential, which we call therandom walk representation condition, which allows us to obtain a strictly convex Hamiltonian, and implies the random walk representation, permitting us to apply the techniques of Helffer and Sjöstrand [23] or Deuschel [13]. The random walk representation condition, and implicitly the strict convexity of the new Hamiltonian, can be verified under our assumptions as in (A0), (A1) and (A2). Note that the method in [8] is more general and could be applied to non-bipartite graphs.

A natural question to ask is whether we can iterate the coarse graining procedure in our current paper and find a scheme which could possibly lower the temperature towards the criticalβ_c, which marks the transition from a unique gradient Gibbs measure μ(as proved in Theorem 1.8in our paper for arbitrary tiltu) to multiple gradient Gibbs measuresμ(as proved in [2] for tiltu=0). Note that iterating the coarse graining scheme is an interesting open problem. One of the main difficulties is that, after iteration, the bond structure on the even sites ofZ^d changes, and we no longer have a bi-partite graph. Currently, we could use our method as detailed in Sections2 and3, to keep integrating out lattice points so that the new Hamiltonian at each step, always of gradient type, can be separated into a strictly convex part and a non-convex perturbation; however, at this point, our technique for estimation of covariances as given in Section3, is not robust enough to allow us to keep coarse graining the lattice points for more than a finite number of steps, before we stop being able to improve the assumptions on our initial perturbationg.

The rest of the paper is organized as follows: In Section2 we present the odd/even characterization of the gra- dient field. In Section3 we give the formulation of the random walk representation condition, which is verified in Theorem3.4under conditions (A0), (A1) and (A2). Section3also presents a few examples, in particular we show that our criteria gets close to the Biskup–Kotecký phase co-existence regime, both for the case of the zero and the non-zero tiltu(see example (a) in Section3.2). In Section4we prove Theorem1.8, our main result on uniqueness of ergodic gradient Gibbs measure with given tiltu, which is based on adaptations of [19], assuming the random walk representation condition. Section5deals with the decay of covariances and the proof is based on the random walk representation for the field at the even sites which allows us to use the result of [11]. Section6 shows the central limit theorem, here again we focus on the field at even sites and apply the random walk representation idea of [22].

Section7proves the strict convexity of the surface tension, or free energy, which follows from the convexity of the Hamiltonian for the gradient field restricted to the even sites. Finally, theAppendixprovides explicit computations for our one-step coarse graining procedure in the special case of potentials considered by [2] (see also example (a) in Section3.2).

2. Even/odd representation

There are two key results in this section. The first one is Lemma2.10, where we are restricting the height differences to the even sites, which induces a∇φmeasure on the even lattice with a different bond structure. The second main result of this section is Lemma2.11, where we give a formula for the conditional of a∇φ-Gibbs measure on the height differences between even sites. These two results will be essential for the proof for one of our main results, that is for the proof of the uniqueness of ergodic component of Theorem1.8.

In Section2.1we introduce the notation for the bond variables on the even subset ofZ^d, in Section2.2we define theφ-Gibbs measure and the∇φ-Gibbs measure corresponding to the even subset ofZ^dand in Section2.3we present the relationship between the∇φ-Gibbs measures for the bonds onZ^dand the∇φfor the bonds on even subset ofZ^d, when their corresponding finite volumeφ-Gibbs measures are related by restriction.

2.1. Notation for the bond variables on the even subset ofZ^d

AsZ^dis a bipartite graph, we will label the vertices ofZ^d asevenandoddvertices, such that everyevenvertex has onlyoddnearest neighbor vertices and vice-versa.

Let Z^dev:=

a=(a₁, a₂, . . . , a_d)∈Z^{d d}

i=1

a_i=2p, p∈Z

and

Z^d_od:=

a=(a₁, a₂, . . . , a_d)∈Z^{d d}

i=1

a_i=2p+1, p∈Z

.

(See also Fig. 1.)

Fig. 1. The bonds of 0 inZ²_ev.

LetΛev⊂Z^devfinite. We will next define the bonds inZ^devin a similar fashion to the definitions for bonds onZ^d. Let Z^dev

_∗ :=

b=(x_b, y_b)|x_b, y_b∈Z^dev,x_b−y_b =2, bdirected fromx_btoy_b , (Λev)^∗:=

Z^dev

_∗

∩(Λev×Λev), (Λev)^∗:=

b=(x_b, y_b)∈ Z^dev

_∗

|x_b∈Λevory_b∈Λev

,

∂(Λ_ev)^∗:=

b=(x_b, y_b)|x_b∈Z^d_ev\Λ_ev, y_b∈Λ_ev,x_b−y_b =2 and

∂Λ_ev:=

y∈Z^d_ev\Λ_ev|y−x =2 for somex∈Λ_ev .

Note that throughout the rest of the paper, we will refer to the bonds on(Z^dev)^∗as theeven bonds.

Aneven plaquetteis a closed loopAev= {b⁽¹⁾, b⁽²⁾, . . . , b⁽ⁿ⁾}, whereb⁽ⁱ⁾∈(Z^dev)^∗,n∈ {3,4}, such that{x_b(i), i= 1, . . . , n}consists ofn different points in Z^d_ev. The field η= {η(b)} ∈R^(Zdev)∗ is said to satisfy the even plaquette conditionif

η(b)= −η(−b) for allb∈ Z^dev

_∗

and

b∈Aev

η(b)=0 for allevenplaquettes inZ^dev. (14)

Letχev be the set of all η∈R^(Zdev)∗ which satisfy the even plaquette condition. For eachb=(xb, yb)∈(Z^dev)^∗we define theeven height differencesη_ev(b):= ∇evφ(b)=φ(y_b)−φ(x_b). The heightsφ^{ηev,φ (0)}can be constructed from the height differencesη_evand the height variableφ(0)atx=0 as

φ^{ηev,φ (0)}(x):=

b∈C_0,x^ev

ηev(b)+φ(0), (15)

wherex∈Z^devandC_0,x^ev is an arbitrary path inZ^dev connecting 0 andx. Note thatφ^{η,φ (0)}(x)is well-defined ifηev= {η_ev(b)} ∈χ_ev. We also define χ_ev,r similarly as we define χ_r. As onZ^d, letP (χ_ev)be the set of all probability measures onχev and letP2(χev)be thoseμ∈P (χev)satisfyingE^μ[|ηev(b)|²]<∞for eachb∈(Z^dev)^∗. We denote χev,r=χ∩L²_r equipped with the norm| · |r.

Remark 2.1. Letη∈χ.Using the plaquette condition property ofη,we will defineηev,the induced bond variables on the even lattice,fromηthus:ifb1=(x, x+ei),b2=(x+ej, x)andbev=(x+ej, x+ei),we defineηev(bev)= η(b₁)+η(b₂).Note thatη_ev∈χ_ev.

Remark 2.2. Throughout the rest of the paper,we will use the notationφev, ψeveither for a stand alone configuration on the even vertices,or in relation to the restriction ofφto the even vertices.ηev, ξevwill denote configurations on the

even bonds.Similarly,Λ_evwill either be a stand alone subset ofZ^devor will be used in relation to the restriction of a setΛ⊂Z^dtoZ^dev.ForΛ⊂Z^d,we will denoteΛ_od:=Z^d_od∩Λ.

2.2. Definition of∇φ-Gibbs measure on(Z^dev)^∗

For everyξev∈χevand finiteΛev⊂Z^dev, the space of all possible configurations of height differences on(Λev)^∗for given boundary conditionξevis defined as

χ_(Λ

ev)^∗,ξev= ηev=

ηev(b)

b∈(Λev)^∗, ηev∨ξev∈χev

,

whereηev∨ξev∈χevis determined by(ηev∨ξev)(b)=ηev(b)forb∈(Λev)^∗and=ξev(b)forb /∈(Λev)^∗.

Theφ-Gibbs measureν^ev onZ^dev and the∇φ-Gibbs measureμ^ev on(Z^dev)^∗with given HamiltonianH^ev can be defined similarly to theφ-Gibbs measure and the∇φ-Gibbs measure in Sections2.1and 1.2.2. They are basically a φ-Gibbs and∇φ-Gibbs measure on a different graph, with vertex and edge sets(Z^dev, (Z^dev)^∗). They are defined via the corresponding HamiltonianH_Λ^ev

ev,ξev, assumed ofevengradient type, via thefinite volume Gibbs measureν_Λ^ev

ev,ψev

onZ^devand thefinite volume∇-Gibbs measureμ^ev_Λ

ev,ψ_evon(Z^dev)^∗. Let

H^ev:=

H_Λ^ev

ev,ξev

Λ_ev⊂Z^dev,ξ_ev∈χ_ev

and let Gev

H^ev :=

μev∈P2(χev): μ^evis∇φ-Gibbs measure on Z^dev

_∗

with given HamiltonianH^ev . Remark 2.3. Similar to Remark 1.2, when Z^dev \ Λ_ev is connected, χ_(Λ

ev)^∗,ξev is an affine space such that dimχ_(Λ

ev)^∗,ξev = |Λev|.Fixing a pointx0∈/Λev,we consider the mapJ_Λ^ev,ξ

ev :χev→R^Zdev,such thatηev→ {φev(x)}, with

φ(x):=

b∈C_x^ev

0,x

(ηev∨ξev)(b), x∈Λev,

for a chainC_x^ev

0,x connectingx₀andxand for fixedφ(x₀), φ(x):=ψ^{ξev,φ (x0)}(x)=

b∈Cx0,x

ξ_ev(b)+φ(x₀), x /∈Λ_ev.

Remark 2.4. For everyξ_ev∈χ_ev anda∈R,letψ_ev=φ^ξev,a be defined by(15)and consider the measureν_Λev,ψev. Thenμ_Λev,ψev is the image measure ofν_Λev,ψev under the map{φ(x)}x∈Λev → {η_ev(b):= ∇(φ_ev∨ψ_ev)(b)}_b∈(Λev)∗. Note that the image measure is determined only byξ_evand is independent of the choice ofa.

2.3. Induced∇φ-Gibbs measure on(Z^dev)^∗

Throughout this section, we will make the following notation conventions. For φ, ψ ∈R^Zd, we define φ_ev :=

(φ(x))_x∈Zd

ev, ψ_ev:=(ψ (x))_x∈Zd

ev. Forη, ξ ∈χ, we defineη_evandξ_evaccording to Remark2.1.

Definition 2.5. LetΛevbe a finite set inZ^dev.We construct a finite setΛ⊂Z^dassociated toΛevas follows:ifx∈Λev, thenx∈Λandx+ei∈Λfor alli∈I= {−n,−n+1, . . . , n} \ {0}.Note that by definition,∂Λ=∂Λev,where the boundary operations are performed in the graphs(Z^d, (Z^d)^∗)and(Z^dev, (Z^dev)^∗),respectively(see Figs2and3).

Lemma 2.6 (Induced finite volume φ-Gibbs measure on Z^dev). Let Λev ⊂Z^dev and letΛ be the associated set inZ^d,as defined in Definition 2.5.Let νΛ,ψ be the finite volume Gibbs measure on Λ with boundary condition

Fig. 2. The graph ofΛev.

Fig. 3. The graph ofΛassociated toΛev.

ψ and with Hamiltonian H_Λ,ψ defined as in(3). We define the induced finite volume Gibbs measure onZ^d_ev as ν_Λ^ev

ev,ψev:=νΛ,ψ|_F(Z^dev).Thenν_Λ^ev

ev,ψevhas HamiltonianH_Λ^ev

ev,ψev,where H_Λ^ev

ev,ψev(φev):=

x∈Λod

F_x

φ(x+e_i)

i∈I

with

(16) F_x

φ(x+e_i)

i∈I

= −log

Re^{−2βi∈IU (∇iφ (x))}dφ(x).

Remark 2.7. Note that for any constantC∈R,by using the change of variablesφ(x)→φ(x)+C in the integral formula forFx((φ(x+ei))i∈I)in(16),we have

F_x

φ(x+e_i)

i∈I

=F_x

φ(x+e_i)+C

i∈I

.

In particular,this means that for any fixedk∈I Fx

φ(x+ei)

i∈I

=Fx

φ(x+ei)−φ(x+ek)

i∈I

. (17)

Therefore we are still dealing with a gradient system.However,it is in general no longer a two-body gradient system.

Fx((φ(x+ei))i∈I),and consequentlyH_Λ^ev

ev,ψev,are functions of the even gradients by(16)and(17).