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Stochastic Interface Models

Tadahisa Funaki

Graduate School of Mathematical Sciences, The University of Tokyo, Komaba,Tokyo 153-8914, JAPAN [email protected]

Summary. In these notes we try to review developments in the last decade of thetheory on stochastic models for interfaces arising in two phase system, mostly on theso-called ∇ϕ interface model. We are, in particular, interested in the scaling limitswhich pass from the microscopic models to macroscopic level. Such limit proceduresare formulated as classical limit theorems in probability theory such as the law oflarge numbers, the central limit theorem and the large deviation principles.

Key words: Random interfaces, Effective interfaces, Phase coexistence and sepa-ration, Ginzburg-Landau model, Massless model, Random walk representation, Sur-face tension, Wulff shape, Hydrodynamic limit, Motion by mean curvature, Evolu-tionary variational inequality, Fluctuations, Large deviations, Free boundaries.

2000 Mathematics Subject Classification: 60-02 (60K35, 60H30, 60H15), 82-02(82B24, 82B31, 82B41, 82C24, 82C31, 82C41), 35J20, 35K55, 35R35

2 T. Funaki

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2 Quick overview of the results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.3 Derivation of effective interface models from Ising model . . . . . . . . . 81.4 Basic notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2 ∇ϕ interface model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.1 Height variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.2 Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.3 Equilibrium states (Gibbs measures) . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.4 Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.5 Scaling limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.6 Quadratic potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3 Gaussian equilibrium systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.1 Gaussian systems in a finite region . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.2 Gaussian systems on Zd . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.3 Massive Gaussian systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303.4 Macroscopic scaling limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

4 Random walk representation and fundamental inequalities 39

4.1 Helffer-Sjöstrand representation and FKG inequality . . . . . . . . . . . . . 394.2 Brascamp-Lieb inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434.3 Estimates of Nash-Aronson’s type and long correlation . . . . . . . . . . . 454.4 Thermodynamic limit and construction of ∇ϕ-Gibbs measures . . . . 484.5 Construction of ϕ-Gibbs measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

5 Surface tension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

5.1 Definition of surface tension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 515.2 Quadratic potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535.3 Fundamental properties of surface tension . . . . . . . . . . . . . . . . . . . . . . 545.4 Proof of Theorems 5.3and 5.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 555.5 Surface tension in one dimensional systems . . . . . . . . . . . . . . . . . . . . . 59

6 Large deviation and concentration properties . . . . . . . . . . . . . 62

6.1 LDP with weak self potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 636.2 Concentration properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 666.3 LDP with weak self potentials in one dimension . . . . . . . . . . . . . . . . . 706.4 LDP for δ-pinning in one dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . 786.5 Outline of the proof of Theorem 6.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 806.6 Critical LDP. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

7 Entropic repulsion, pinning and wetting transition . . . . . . . . 89

Stochastic Interface Models 3

7.1 Entropic repulsion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 897.2 Pinning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 947.3 Wetting transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

8 Central limit theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

9 Characterization of ∇ϕ-Gibbs measures . . . . . . . . . . . . . . . . . . . 106

9.1 ϕ-dynamics on Zd and ∇ϕ-dynamics on (Zd)∗ . . . . . . . . . . . . . . . . . . 1069.2 Stationary measures and ∇ϕ-Gibbs measures . . . . . . . . . . . . . . . . . . . 1089.3 Proof of Theorem 9.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1099.4 Proof of Proposition 9.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1149.5 Uniqueness of ϕ-Gibbs measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

10 Hydrodynamic limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

10.1 Space-time diffusive scaling limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11710.2 The nonlinear PDE (10.4) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12110.3 Local equilibria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12410.4 Proof of Theorem 10.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13010.5 Surface diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

11 Equilibrium fluctuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

12 Dynamic large deviation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

12.1 Dynamic LDP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13812.2 Dynamic rate functional . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13812.3 Relation to the static LDP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

13 Hydrodynamic limit on a wall . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

13.1 Dynamics on a wall . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14113.2 Hydrodynamic limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

14 Equilibrium fluctuation on a wall and entropic repulsion . . 144

14.1 The case attached to the wall . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14414.2 The case away from the wall . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14514.3 Dynamic entropic repulsion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

15 Dynamics in two media and pinning dynamics on a wall . . . 147

15.1 Dynamics in two media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14715.2 Pinning dynamics on a wall . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

16 Other dynamic models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

16.1 Stochastic lattice gas and free boundary problems . . . . . . . . . . . . . . . 15316.2 Interacting Brownian particles at zero temperature . . . . . . . . . . . . . . 15416.3 Singular limits for stochastic reaction-diffusion equations . . . . . . . . . 15816.4 Limit shape of random Young diagrams . . . . . . . . . . . . . . . . . . . . . . . . 16216.5 Growing interfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

4 T. Funaki

1 Introduction

1.1 Background

The water changes its state to ice or vapor together with variations in tem-perature. Each of these three states (liquid/solid/gas) is macroscopically ho-mogeneous and called a phase (or a pure phase) in physics. The water and theice can coexist at temperature 0

◦

C. In fact, under various physical situationsespecially at low temperature, more than one distinct pure phases coexist inspace and different phases are separated by fairly sharp hypersurfaces calledinterfaces. Snow crystals in the vapor or alloys consisting of two types ofmetals are typical examples. Crystals are macroscopic objects, which haveordered arrangements of atoms or molecules in microscopic scale.

Wulff [254] in 1901 proposed a variational principle, at thermodynamiclevel or from the phenomenological point of view, for determining the shapeof interfaces for crystals. Let E ⊂ Rd be a crystal shape. Its boundary ∂E isthen an interface and an energy called the total surface tension is associatedwith each interface by

W(E) =∫

∂E

σ(n(x)) dx, (1.1)

where σ = σ(n) ≥ 0 is the surface tension of flat hyperplane in Rd with unitnormal vector n ∈ Sd−1 and dx represents the volume element on ∂E. Theinterface has locally an energy σ(n(x)) depending on its tilt n = n(x) and,integrating it over the surface ∂E, the Wulff functional W(E) is defined. Foran alloy consisting of two types of metals A and B, E is the region occupiedby A-type’s metal so that its volume is always kept invariant if the amountof each metal is fixed.

It is expected that the interface, which is in equilibrium and stable, mini-mizes its total energy and this naturally leads us to the variational problem:

minvol (E)=v

W(E) (1.2)

under the condition that the total volume of the crystal E (e.g., the region oc-cupied by A-type’s metal) is fixed to be v > 0 . The minimizer E of (1.2) andits explicit geometric expression are called the Wulff shape and the Wulffconstruction, respectively. Especially when the surface tension σ is indepen-dent of the direction n, W(E) coincides with the surface area of ∂E (exceptconstant multipliers) and (1.2) is equivalent to the well-known isoperimet-ric problem. It is one of quite general and fundamental principles in physicsthat physically realizable phenomena might be characterized by variationalprinciples. Wulff’s variational problem is one of the typical examples.

Crystals are, as we have already pointed out, macroscopic objects. It is aprincipal goal of statistical mechanics to understand such macroscopic phe-nomena in nature from microscopic level of atoms or molecules. Dobrushin,


Kotecký and Shlosman [86] studied the Wulff’s problem from microscopicpoint of view for the first time. They have employed the ferromagnetic Isingmodel as a microscopic model and established, at sufficiently low tempera-tures, the large deviation principle for the sequence of corresponding Gibbsmeasures on finite domains when the volumes of these domains diverge toinfinity. It was shown that the large deviation rate functional is exactly theWulff functional W(E) with the surface tension σ(n) determined thermody-namically from the underlying Gibbs measures. As a consequence, under thecanonical Gibbs measures obtained by conditioning the macroscopic volumeoccupied by + spins to be constant, a law of large numbers is proved andthe Wulff shape is obtained in the limit. The results of Dobrushin et al. wereafterward generalized by Ioffe and Schonmann [152], Bodineau [20], Cerf andPisztora [52] and others; see a review paper [22].

Once an equilibrium situation is understood to some extent, the next targetis obviously the analysis of the corresponding dynamics. The situation thattwo distinct pure phases coexist and are separated by a sharp interface willpersist under the time evolution and the interface will relax slowly. The goalis to investigate the motion of interface on a properly chosen coarse space-time scale. The time evolution corresponding to the Ising model is a reversiblespin-flip dynamics, the so-called Glauber or Kawasaki dynamics which maybe the prime examples. Spin at each site randomly flips and changes its signunder the dynamics without or with conservation law. At sufficiently lowtemperatures, the interactions between spins on two neighboring sites becomestrong enough to incline them to have the common signs with high probabilityand most changes occur near the interface. The shape of interface is howeverrather complicated; for instance, it has overhangs or bubbles.

A class of effective interface models is introduced by avoiding such compli-cations and directly modeling the interface degree of freedom at microscopiclevel; see Sect. 1.3. These models are, at one side, compromises between thedescription of physical phenomena and mathematical requirements but, onthe other side, explain the phenomena in satisfactory good way. The aim ofthese notes is to try to give an overview of results mostly on the ∇ϕ interfacemodel, which is one of such effective interface models.

As we have observed, in statistical mechanics, there are at least two dif-ferent scales: macroscopic and microscopic ones. The procedures connectingmicroscopic models with the macroscopic phenomena are realized by takingthe scaling limits. The scaling parameter N ∈ N represents the ratio of themacroscopically typical length (e.g., 1 cm) to the microscopic one (e.g., 1 nm)and it is finite, but turns out to be quite large (N = 107 in this example).The physical phenomena can be mathematically understood only by takingthe limit N → ∞. The dynamics involves the scalings also in time. Within amacroscopic unit length of time, the molecules collide with each other withtremendous frequency. Since the microscopic models such as the Ising modeland the ∇ϕ interface model involve randomness, the limit procedure N → ∞

6 T. Funaki

can be formulated in the framework of classical limit theorems in probabilitytheory.

The principal ideas behind these limit theorems are that, by the ergodicor mixing properties of the microscopic systems, the microscopic (physical)quantities are locally in macroscopic scale averaged or homogenized under thescaling limits. The macroscopic observables are obtained under such averagingeffects. However, the ∇ϕ interface model which we shall discuss in the presentnotes has only an extremely weak mixing property and this sometimes makesthe analysis of the model difficult. For instance, the thermodynamic quantitymay diverge under the usual scaling. This suggests the necessity of introducingscalings different from the usual one to obtain a nontrivial limit.

1.2 Quick overview of the results

In Sect. 2, the ∇ϕ interface model is precisely introduced. The basic micro-scopic objects are height variables φ of interfaces. Assigning an energy H(φ)to each height variable, its statistical ensemble in equilibrium is defined by theGibbs measures. Then, the corresponding time evolution called the Ginzburg-Landau ∇ϕ interface model is constructed in such a way that it is reversibleunder the Gibbs measures, in other words, the detailed balance is fulfilled. Thescaling limits connecting microscopic and macroscopic levels will be explained.

The ∇ϕ interface model with quadratic potentials is discussed in Sect. 3 asa warming up before studying general case with convex potentials. In Sect. 4,fundamental tools like Helffer-Sjöstrand (random walk) representation, FKGinequality and Brascamp-Lieb inequality are presented.

A basic role in various limit theorems is played by the so-called surfacetension σ(u), u ∈ Rd. The function σ is a macroscopic or thermodynamicfunction and will be introduced in Sect. 5. The limit theorems under thescalings can be formulated in the terminology of probability theory as follows:

Law of large numbers (LLN): Macroscopic quantity obtained under thescaling limit from randomly fluctuating microscopic objects, i.e., heightvariables of interfaces in our model, becomes deterministic due to certainaveraging effects.

Central limit theorem (CLT): Fluctuations around the deterministiclimit are studied.

Large deviation principle (LDP): LDPs for macroscopically scaledheight variables are sometimes useful to show the LLNs.

From the physical point of view, these limit theorems are classified into twotypes: static results on the equilibrium Gibbs measures and dynamic results:

(1) Static results, Sects. 6-9.

LDP, LLN and derivation of variational principles (VP), Sect. 6:LDP was studied for Gaussian case by Ben Arous and Deuschel [12] andfor general Gibbsian case by Deuschel, Giacomin and Ioffe [77]. For height


variables conditioned to be positive and to have definite total volume, theshape of most probable droplet called the Wulff shape is determined asa minimizer of the total surface tension as a consequence of LDP. Addingan effect of weak self potentials to the system, Funaki and Sakagawa [123]derived the VPs of Alt and Caffarelli [5] or Alt, Caffarelli and Friedman [6].Bolthausen and Ioffe [31] discussed under additional pinning effect at awall for 2+1 dimensional system and obtained the Winterbottom shapein the limit.

Entropic repulsion (wall effect), Sect. 7.1: The entropic repulsion isthe problem to study, when a hard wall is settled at the height level0, how high the interfaces are pushed up by the randomness (i.e., theentropic effect) naturally existing in the Gibbs measures. The problem wasposed by Lebowitz and Maes [186] and then investigated by Bolthausen,Deuschel and Zeitouni [29], Deuschel [74], Deuschel and Giacomin [75] forGaussian case and by Deuschel and Giacomin [76] for general Gibbsiancase.

Pinning and wetting transition, Sects. 7.2, 7.3: The pinning is the prob-lem to study, under the effect of weak force attracting interfaces to theheight level 0, whether the field is really localized or not. The problemwas discussed by Dunlop, Magnen, Rivasseau and Roche [93], Deuscheland Velenik [81], Ioffe and Velenik [153] and Bolthausen and Velenik [32].The two effects of entropic repulsion and pinning conflict with each other,and a natural question to be addressed is which effect is dominant inthe system. In one and two dimensions, a phase transition called wettingtransition occurs. This fact was first observed by Fisher [101] in one di-mension, followed by Bolthausen, Deuschel and Zeitouni [30] and Caputoand Velenik [49].

CLT, Sect. 8: Naddaf and Spencer [202] investigated CLT for Gibbs mea-sures. The result is nontrivial since the Gibbs measures have long corre-lations.

Characterization of ∇ϕ-Gibbs measures, Definition 2.2, Sect. 9: Thefamily of all (tempered and shift invariant) ∇ϕ-Gibbs measures is charac-terized based on the coupling argument for the corresponding dynamics.This result plays a key role in the proof of the hydrodynamic limit.

(2) Dynamic results I, Sects. 10-12.

Hydrodynamic limit (LLN) and derivation of motion by mean cur-vature with anisotropy, Sect. 10: LLN is shown under the time evolu-tion. This procedure is called the hydrodynamic limit and established byFunaki and Spohn [124]. Motion by mean curvature (MMC) except forsome anisotropy is derived in the limit. The diffusion matrix of the limitequation is formally given by Hessian of the surface tension.

Equilibrium fluctuation (CLT), Sect. 11: Dynamic CLT in equilibrium isstudied and an infinite dimensional Ornstein-Uhlenbeck process is derived

8 T. Funaki

in the limit by Giacomin, Olla and Spohn [135]. The identification ofthe covariance matrix with Hessian of the surface tension, however, stillremains open.

LDP, Sect. 12: Dynamic LDP was discussed by Funaki and Nishikawa [121].

(3) Dynamic results II, Sects. 13-15.

The dynamics under the effects of wall or additional weak self potentialsis studied.

Hydrodynamic limit on a wall, Sect. 13: The limit is MMC with reflec-tion and described by an evolutionary variational inequality, Funaki[117].

Equilibrium fluctuation (CLT) on a wall, Sects. 14.1, 14.2: A stochas-tic PDE with reflection is obtained under the scaling limit, Funaki andOlla [122].

Dynamic entropic repulsion, Sect. 14.3: The problem of entropic repul-sion is investigated under the dynamics, Deuschel and Nishikawa [80] andothers.

Dynamics in two media, Sect. 15.1: The dynamics associated with theHamiltonian added a weak self potential is discussed.

Pinning dynamics on a wall, Sect. 15.2: Dynamics under the effects ofboth pinning and repulsion is constructed.

(4) Other dynamic models for interfaces, Sect. 16.

The following five topics are discussed in the last supplementary section.

Stochastic lattice gas and free boundary problemsInteracting Brownian particles at zero temperatureSingular limits for stochastic reaction-diffusion equationsLimit shape of random Young diagramsGrowing interfaces

Funaki [116] and Giacomin [130], [132], [133] are survey papers on the∇ϕ interface model. See also [125], [210], [224] for problems on interfaces andcrystals.

1.3 Derivation of effective interface models from Ising model

Let us briefly and rather formally explain how one can derive the effectiveinterface models from the ferromagnetic Ising model at sufficiently low tem-perature. In the Ising model, the energy is associated to each ± spin configu-ration s = {s(x);x ∈ Λ`} ∈ {+1,−1}Λ` on a large box Λ` := [−`, `]d ∩ Zd asthe sum over all bonds 〈x, y〉 in Λ` (i.e., x, y ∈ Λ` : |x− y| = 1)

H(s) = −∑

〈x,y〉⊂Λ`

s(x)s(y).


The sum is usually defined under certain boundary conditions. We shall con-sider, for simplicity, only when d = 2. The function H(s) can be rewrittenas

H(s) = 2|γ| (+ constant)in terms of the set of contours γ = γ(s) on the dual lattice corresponding tos, which separate two regions consisting of sites occupied by + and − spins,respectively, where |γ| denotes the number of bonds in γ (the total length ofγ) and an additional constant in H(s) is independent of the configurations s.Under the Gibbs measure

µ`(s) =1

Z`e−βH(s), s ∈ {+1,−1}Λ`,

with the normalization constant Z`, if the temperature T (β = 1/kT , k > 0 isthe Boltzmann constant) is sufficiently low, the configurations of spins whichhave the same values on neighboring sites overwhelm the probability, sincesuch configurations have smaller energies. In other words, when there is asingle large contour γ, the probability that the configurations in Fig. 2 havingbubbles arise is very little and almost negligible. We can therefore disregard

Fig. 1. Possible configurations. Fig. 2. Neglected configurations.

(with high probability) the configurations with bubbles and assume that theconfigurations like in Fig. 1 can only appear. Such spin configurations s areequivalently represented by the height variables φ = {φ(x) ∈ [−`, `] ∩ Z;x ∈[−`, `]d−1∩Zd−1} (in fact, we are considering the case of d = 2) which measurethe distances of γ from the x-axis, one fixed hyperplane. Then, the energyH(s)has another form

H(φ) = 2∑

〈x,y〉⊂[−`,`]d−1∩Zd−1|φ(x) − φ(y)| (1.3)

10 T. Funaki

up to an additional constant; notice that the number of horizontal bonds inγ is always fixed. The model for random interfaces φ : [−`, `]d−1 ∩ Zd−1 → Zwith the energy (1.3) is called the SOS (Solid on Solid) model. One canfurther replace the space Z for values of height variables with continuum Rand |φ(x)−φ(y)| with V (φ(x)−φ(y)), and this leads us to the ∇ϕ interfacemodel. As a generalization of the function V (η) = |η|, it is natural to supposethat the potential function V is convex and symmetric (even) so that theenergy is small when the differences of heights φ : [−`, `]d−1 ∩ Zd−1 → R onneighboring sites are small, in other words, when the interfaces are more flat.

1.4 Basic notation

• For Λ ⊂ Zd (d dimensional square lattice),

∂+Λ = {x /∈ Λ; |x− y| = 1 for some y ∈ Λ}

is the outer boundary of Λ and Λ = Λ∪∂+Λ is the closure of Λ, respectively,where x /∈ Λ means x ∈ Λc = Zd \ Λ. The inner boundary of Λ is

∂−Λ = {x ∈ Λ; |x− y| = 1 for some y /∈ Λ}.

• Λ b Zd means that Λ is a finite subset of Zd: |Λ|(= ]Λ)


people, in particular, with J.-D. Deuschel, G. Giacomin, D. Ioffe, S. Olla, G.S.Weiss and N. Yoshida. H. Sakagawa read an early version in part and gave meseveral suggestions for improvement. Professor J. Picard invited me to delivera series of lectures at the International Probability School at Saint-Flour,2003. I deeply thank all of these people.

12 T. Funaki

2 ∇ϕ interface model

The ∇ϕ interface model has a rather simplified feature, for example, whenit is compared with the Ising model, as we have pointed out. It is, however,equipped with a sufficiently wide variety of nontrivial aspects and serves as auseful model to explain physical behavior of interfaces from microscopic pointof view. In this section we introduce the model.

2.1 Height variables

We are concerned with a hypersurface (interface) embedded in d + 1 dimen-sional space Rd+1, which separates two distinct pure phases. Notice that, inSect. 1.3, we discussed in d dimensional space; however, here and after d isreplaced with d+1. To avoid complications, we assume that the interface hasno overhangs nor bubbles and accordingly that it is represented as a graphviewed from a certain d dimensional fixed reference hyperplane Γ located inthe space Rd+1. In other words, the location of the interface is described by theheight variables φ = {φ(x) ∈ R;x ∈ Γ}, which measure the vertical distancesbetween the interface and Γ . The variables φ are microscopic objects, and thespace Γ is discretized and taken as Γ = Λ(b Zd), in particular, Γ = DN witha (macroscopic) bounded domain D in Rd or lattice torus TdN or Z

d. Here Nrepresents the size of the microscopic system, and our main interest will bein analyzing the asymptotic behavior of the system under the scaling limitN → ∞.

2.2 Hamiltonian

An energy is associated with each height variable φ : Γ → R by assigningpenalty according to its tilts (slopes). Namely, we define the HamiltonianH(φ) as the sum over all bonds (i.e., pairs of nearest neighbor sites) 〈x, y〉 inΓ when Γ = TdN or Z

d, and in Γ when Γ = DN or Γ = Λ b Zd in general

H(φ) ≡ HψΓ (φ) =∑

〈x,y〉⊂Γ (or Γ )

V (φ(x) − φ(y)). (2.1)

Note that the boundary conditions ψ = {ψ(x);x ∈ ∂+Γ} are required todefine the sum (2.1) for Γ = DN , i.e., we assume

φ(x) = ψ(x), x ∈ ∂+Γ,

in the sum. When Γ = Zd, (2.1) is a formal infinite sum. The (interaction) po-tential V is smooth, symmetric and strictly convex. More precisely, through-out the present notes we require the following three conditions on the potentialV = V (η):


(V1) (smoothness) V ∈ C2(R),(V2) (symmetry) V (−η) = V (η), η ∈ R, (2.2)(V3) (strict convexity) c− ≤ V ′′(η) ≤ c+, η ∈ R, for some c−, c+ > 0.

The surface φ has low energy if the tilts |φ(x) − φ(y)| are small. The energy(2.1) of the interface φ is constructed in such a manner that it is invariantunder a uniform translation φ(x) → φ(x) + h for all x ∈ Zd (or x ∈ Γ ) andh ∈ R. A typical example of V satisfying the conditions (2.2) is a quadraticpotential V (η) = c2η

2, c > 0.For every Λ ⊂ Zd, Λ∗ denotes the set of all directed bonds b = 〈x, y〉 in Λ,

which are directed from y to x. We write xb = x, yb = y for b = 〈x, y〉. Foreach b ∈ (Zd)∗ and φ = {φ(x);x ∈ Zd} ∈ RZd , define

∇φ(b) = φ(xb) − φ(yb).

We also define ∇iφ(x) = φ(x + ei) − φ(x), 1 ≤ i ≤ d for x ∈ Zdwhere ei ∈ Zd is the i-th unit vector given by (ei)j = δij . The variables∇φ(x) = {∇iφ(x)}1≤i≤d ∈ Rd represent vector field of height differencesor sometimes called tilt (or gradients) of φ. The Hamiltonian H(φ) is thenrewritten as

H(φ) =1

2

∑

b∈Γ∗(or Γ∗)

V (∇φ(b)). (2.3)

The factor 1/2 is needed because each undirected bond b = 〈x, y〉 is countedtwice in the sum. Since the energy is determined from the height differences∇φ, the model is called the ∇ϕ interface model.

Remark 2.1 (1) The sum (2.1) is meaningful only when the potential V issymmetric, while the expression (2.3) makes sense for asymmetric V . How-ever, note that the sum (2.3) is essentially invariant (except for the boundarycontributions) if V is replaced with its symmetrization 12{V (η) + V (−η)}.(2) The potential V can be generalized to the bond-dependent case: {Vb =Vb(η); b ∈ (Zd)∗} so that the corresponding Hamiltonian is defined by (2.3)with V replaced by Vb; see Example 5.3, Problem 10.1 below and [230]. Thisformulation truly covers the asymmetric potentials.

Remark 2.2 (1) In the quantum field theory, H is called massless Hamil-tonian and well studied in ’80s. Massive Hamiltonian is given by Hm(φ) =H(φ) + 12m

2∑

x φ(x)2,m > 0. The Hamiltonian with weak self potentials or

pinning potentials will be introduced in Sect. 6.1 or in Sect. 6.4 (see also Sect.7.2), respectively.(2) In our model, height variables φ(x) themselves are not discretized. TheSOS (solid on solid) model is a model obtained discretizing the height vari-ables simultaneously: φ(x) ∈ Z+ and with V (η) = |η|, cf. Sect. 1.3 and [54],[55], [106].(3) (∆ϕ interface model) In the ∇ϕ interface model, the energy H(φ) is

14 T. Funaki

roughly the surface area of the microscopic interface φ. In fact, this is true forV (η) =

√1 + η2. However, if we are concerned for example with the mem-

brane as the object of our study, its surface area is preserved and alwaysconstant. Therefore the energy should be determined by taking into accountthe next order term like

∑x(∆φ(x))

2, which may be regarded as the curvatureof φ, see [145].

2.3 Equilibrium states (Gibbs measures)

Once the Hamiltonian H is specified, in the formulation of statistical me-chanics, equilibrium states called Gibbs measures can be naturally associatedtaking the effect of random fluctuations into account.

ϕ-Gibbs measures

For a finite region Λ b Zd, the Gibbs measure (more exactly, ϕ-Gibbsmeasure, finite volume ϕ-Gibbs measure or local specification) for the field ofheight variables φ ∈ RΛ over Λ is defined by

µ(dφ) ≡ µψΛ(dφ) =1

ZψΛexp

{−HψΛ (φ)

}dφΛ, (2.4)

with the boundary conditions ψ ∈ R∂+Λ. The term e−HψΛ (φ) is the Boltzmannfactor, while

dφΛ =∏

x∈Λdφ(x)

is the Lebesgue measure on RΛ which represents uniform fluctuations of theinterface. The constant ZψΛ is for normalization defined by

ZψΛ =

∫

RΛ

exp{−HψΛ (φ)

}dφΛ. (2.5)

Note that the conditions (2.2) imply ZψΛ


we regard µψΛ ∈ P(RZd

) in such case. When Γ = TdN , the Gibbs measure

is unnormalizable, since HψΛ (φ) is translation invariant and this makes thenormalization ZTdN = ∞.

For an infinite region Λ : |Λ| = ∞, the expression (2.4) has no meaningsince the Hamiltonian HΛ(φ) is a formal sum. Nevertheless, one can define thenotion of Gibbs measures on Zd based on the well-known DLR formulations.For A ⊂ Zd, we shall denote FA the σ-field of RZ

d

generated by {φ(x);x ∈ A}.

Definition 2.1 The probability measure µ ∈ P(RZd) is called a Gibbs mea-sure for ϕ-field (ϕ-Gibbs measure for short), if its conditional probabilityon FΛc satisfies the DLR equation

µ( · |FΛc)(ψ) = µψΛ( · ), µ-a.e.ψ,

for every Λ b Zd.

It is known that the ϕ-Gibbs measures exist when the dimension d ≥ 3,but not for d = 1, 2. An infinite volume limit (thermodynamic limit) for µ0Λas Λ↗ Zd exists only when d ≥ 3 (cf. Sect. 4.5).

∇ϕ-Gibbs measures

The height variables φ = {φ(x);x ∈ Zd} on Zd automatically determines afield of height differences ∇φ = {∇φ(b); b ∈ (Zd)∗}. One can therefore considerthe distribution µ∇ of ∇ϕ-field under the ϕ-Gibbs measure µ. We shall callµ∇ the ∇ϕ-Gibbs measure. In fact, it is possible to define the ∇ϕ-Gibbsmeasures directly by means of the DLR equations and, in this sense, ∇ϕ-Gibbsmeasures exist for all dimensions d ≥ 1 (cf. Sect. 4.4).

In order to describe the DLR equation for ∇ϕ-Gibbs measures, we firstclarify the structure of the state space for the ∇ϕ-field. It is obvious that theheight variable φ ∈ RZd determines ∇φ ∈ R(Zd)∗ ; however, all η = {η(b)} ∈R

(Zd)∗ can not be the ∇ϕ-field, i.e., it may not be possible to find φ such thatη = ∇φ in general. Indeed, ∇φ always satisfies the loop condition: every sumof ∇φ along a closed loop must vanish. To state more precisely, we introducesome notion.

A sequence of bonds C = {b(1), b(2), . . . , b(n)} is called a chain connectingy and x (y, x ∈ Zd) if yb(1) = y, xb(i) = yb(i+1) for 1 ≤ i ≤ n− 1 and xb(n) = x.The chain C is called a closed loop if xb(n) = yb(1) . A plaquette is a closedloop P = {b(1), b(2), b(3), b(4)} such that {xb(i) , i = 1, .., 4} consists of fourdifferent points. The field η = {η(b)} ∈ R(Zd)∗ is said to satisfy the plaquettecondition if

(P1) η(b) = −η(−b) for all b ∈ (Zd)∗,(P2)

∑

b∈Pη(b) = 0 for all plaquettes P in Zd,

16 T. Funaki

where −b denotes the reversed bond of b. Note that, if φ = {φ(x)} ∈ RZd ,then ∇φ = {∇φ(b)} ∈ R(Zd)∗ automatically satisfies the plaquette condition.The plaquette condition is equivalent to the loop condition:

(L)∑

b∈Cη(b) = 0 for all closed loops C in Zd.

Notice that the condition (P1) follows from (L) by taking the closed loopC = {b,−b}. We set

X = {η ∈ R(Zd)∗ ; η satisfies the loop condition},

then X is the state space for the ∇ϕ-field endowed with the topology inducedfrom the space R(Z

d)∗ having product topology. In fact, the height differences

ηφ ∈ X are associated with the heights φ ∈ RZd by

ηφ(b) := ∇φ(b), b ∈ (Zd)∗, (2.6)

and, conversely, the heights φη,φ(O) ∈ RZd can be constructed from heightdifferences η and the height variable φ(O) at x = O as

φη,φ(O)(x) :=∑

b∈CO,xη(b) + φ(O), (2.7)

where CO,x is an arbitrary chain connecting O and x. Note that φη,φ(O) is

well-defined if η = {η(b)} ∈ X .We next define the finite volume ∇ϕ-Gibbs measures. For every ξ ∈ X

and Λ b Zd the space of all possible configurations of height differences onΛ∗ := {b = 〈x, y〉 ∈ (Zd)∗; x or y ∈ Λ} for given boundary condition ξ isdefined as

XΛ∗,ξ = {η = (η(b))b∈Λ∗ ; η ∨ ξ ∈ X},where η ∨ ξ ∈ X is determined by (η ∨ ξ)(b) = η(b) for b ∈ Λ∗ and = ξ(b) forb /∈ Λ∗. The finite volume ∇ϕ-Gibbs measure in Λ (or, more precisely, in Λ∗)with boundary condition ξ is defined by

µ∇Λ,ξ(dη) =1

ZΛ,ξexp

−1

2

∑

b∈Λ∗V (η(b))

dηΛ,ξ ∈ P(XΛ∗,ξ),

where dηΛ,ξ denotes a uniform measure on the affine space XΛ∗,ξ and ZΛ,ξ isthe normalization constant. We shall sometimes regard µ∇Λ,ξ ∈ P(X ) by con-sidering η(b) = ξ(b) for b /∈ Λ∗ under µ∇Λ,ξ as before. Note that the dimensionof the space XΛ∗,ξ is |Λ| at least if Zd \Λ is connected, since one can associateη with φ = φΛ by

φ(x) =∑

b∈Cx0,x(η ∨ ξ)(b), x ∈ Λ, (2.8)


where x0 /∈ Λ is fixed and Cx0,x is a chain connecting x0 and x.The finite volume ϕ-Gibbs measures and the finite volume ∇ϕ-Gibbs mea-

sures are associated with each other as we have pointed out above. Namely,

given ξ ∈ X and h ∈ R, define ψ ∈ RZd as ψ = φξ,h by (2.7). Then, if φ isµψΛ-distributed with the boundary condition ψ constructed in this way, ∇φ isµ∇Λ,ξ-distributed. The distribution of ∇φ is certainly independent of the choiceof h.

Now, similarly to the definition of the ϕ-Gibbs measures on Zd, one canintroduce the ∇ϕ-Gibbs measures on (Zd)∗.

Definition 2.2 The probability measure µ∇ ∈ P(X ) is called a Gibbs measurefor the height differences (∇ϕ-Gibbs measure for short), if it satisfies theDLR equation

µ∇( · |F(Zd)∗\Λ∗)(ξ) = µ∇Λ,ξ( · ), µ∇-a.e. ξ,

for every Λ b Zd, where F(Zd)∗\Λ∗ stands for the σ-field of X generated by{η(b); b ∈ (Zd)∗ \ Λ∗}.

Markov property

In the Hamiltonian H(φ), the interactions among the height variables areonly counted through the neighboring sites. This structure is reflected as theMarkov property of the field of height variables φ = {φ(x)} under the(finite or infinite volume) ϕ-Gibbs measures µψΛ and µ:

Proposition 2.1 (1) Let Λ b Zd and the boundary condition ψ ∈ R∂+Λbe given. Suppose that Λ is decomposed into three regions A1, A2, B and Bseparates A1 and A2; namely, Λ = A1∪A2∪B, A1∩A2 = A1∩B = A2∩B =∅ and |x1 − x2| > 1 holds for every x1 ∈ A1 and x2 ∈ A2. Then, underthe conditional probability µψΛ ( · |FB), the random variables φA1 and φA2 aremutually independent, where we denote φA1 = {φ(x);x ∈ A1} etc.(2) Let µ ∈ P(RZd) be a ϕ-Gibbs measure. Then, for every A b Zd, therandom variables φA and φAc are mutually independent under the conditionalprobability µ ( · |F∂+A).

In particular, in one dimension, φ = {φ(x)} is a pinned random walkunder µψΛ regarding x as time variables. Let {η(y); y = 1, 2, . . .} be an R-valuedi.i.d. defined on a certain probability space (Ω,P ) having the distributionp(a)da, where

p(a) =1

ze−V (a), a ∈ R

and z =∫

Re−V (a) da is the normalization constant. Then, we have the follow-

ing.

18 T. Funaki

Proposition 2.2 Let Λ = {1, 2, . . . , N−1} ⊂ Z1 and assume that the bound-ary conditions are given by ψ(0) = h0, ψ(N) = h1. Define the height variablesφ = {φ(x);x ∈ Λ}, Λ = {0, 1, 2, . . . , N} by

φ(x) = h0 +

x∑

y=1

η(y), x ∈ Λ,

and consider them under the conditional probability P ( · |φ(N) = h1). Then,φΛ = {φ(x);x ∈ Λ} is µψΛ-distributed.

Shift invariance and ergodicity

Here, we recall the notion of shift invariance and ergodicity under the shifts forϕ-fields and ∇ϕ-fields, respectively, see, e.g., [128]. For x ∈ Zd, we define theshift operators τx : R

Zd → RZd for heights by (τxφ)(y) = φ(y − x) for y ∈ Zd

and φ ∈ RZd . The shifts for height differences are also denoted by τx. Namely,τx : X → X

(or τx : R

(Zd)∗ → R(Zd)∗)

are defined by (τxη)(b) = η(b − x) forb ∈ (Zd)∗ and η ∈ X

(or η ∈ R(Zd)∗

), where b− x = 〈xb − x, yb − x〉 ∈ (Zd)∗.

Definition 2.3 A probability measure µ ∈ P(RZd) is called shift invariantif µ ◦ τ−1x = µ for every x ∈ Zd. A shift invariant µ ∈ P(RZ

d

) is called

ergodic (under the shifts) if {τx}-invariant functions F = F (φ) on RZd

(i.e.,functions satisfying F (τxφ) = F (φ) µ-a.e. for every x ∈ Zd) are constant (µ-a.e.). Similarly, the shift invariance and ergodicity for a probability measure

µ ∈ P(X )(or µ ∈ P(R(Zd)∗)

)are defined.

2.4 Dynamics

Corresponding to the Hamiltonian H(φ), one can naturally introduce a ran-dom time evolution of microscopic height variables φ of the interface. Indeed,we consider the stochastic differential equations (SDEs) for φt = {φt(x);x ∈Γ} ∈ RΓ , t > 0

dφt(x) = −∂H

∂φ(x)(φt)dt+

√2dwt(x), x ∈ Γ, (2.9)

where wt = {wt(x);x ∈ Γ} is a family of independent one dimensional stan-dard Brownian motions. The derivative of H(φ) in the variable φ(x) is givenby

∂H

∂φ(x)(φ) =

∑

y∈Γ (orΓ ):|x−y|=1

V ′(φ(x) − φ(y)), (2.10)

for x ∈ Γ . When Γ b Zd, the SDEs (2.9) have the form


dφt(x) = −∑

y∈Γ :|x−y|=1

V ′(φt(x) − φt(y))dt+√

2dwt(x), x ∈ Γ, (2.11)

subject to the boundary conditions

φt(y) = ψ(y), y ∈ ∂+Γ. (2.12)

When Γ = Zd, although the Hamiltonian H is a formal sum, its derivative(2.10) has an affirmative meaning and we can write down the SDEs for φt =

{φt(x);x ∈ Zd} ∈ RZd

, t > 0

dφt(x) = −∑

y∈Zd:|x−y|=1V ′(φt(x) − φt(y))dt+

√2dwt(x), x ∈ Zd. (2.13)

The SDEs (2.11) with (2.12) or the SDEs (2.13) have unique solutions, sincethe coefficient V ′ in the drift term is Lipschitz continuous by our assump-tions (2.2). For (2.13), since it is an infinite system, one need to introduce aproper function space for solutions, cf. Lemmas 9.1 and 9.2. The evolution ofφt is designed in such a manner that it is stationary and, moreover, reversibleunder the Gibbs measures µψΛ or µ, cf. Proposition 9.4 for the associated ∇ϕ-dynamics. In physical terminology, the equation fulfills the detailed balancecondition. Such evolution or the SDEs are called Ginzburg-Landau dy-namics, distorted Brownian motion or the Langevin equation associatedwith H(φ).

The drift term in the SDEs (2.9) determines the gradient flow along whichthe energy H(φ) decreases. In fact, since the function V is symmetric andconvex, φt(x) > φt(y) implies that −V ′(φt(x) − φt(y)) < 0 so that the driftterm of (2.11) or (2.13) is negative and therefore φt(x) decreases. Converselyif φt(x) < φt(y), the drift is positive and φt(x) increases. Therefore, in bothcases, the drift has an effect to make the interface φ flat. The term

√2wt(x)

gives a random fluctuation which competes against the drift.The Dirichlet form corresponding to the SDEs (2.13) is

E(F,G) ≡ −Eµ[FLG] =∑

x∈ZdEµ[∂F (x, φ)∂G(x, φ)], (2.14)

for F = F (φ), G = G(φ), where Eµ[ · ] denotes the expectation under theGibbs measure µ, L is the generator of the process φt and ∂F (x, φ) :=∂F/∂φ(x), cf. Sects. 4.1 and 10.3. Indeed, at least when Γ b Zd, the genera-tor L of the process φt ∈ RΓ determined by the SDEs (2.9) is the differentialoperator of second order

L =∑

x∈Γ

(∂

∂φ(x)

)2−∑

x∈Γ

∂H

∂φ(x)

∂

∂φ(x)(2.15)

and, by integration by parts formula, we have

20 T. Funaki

∫

RΓ

FLG · e−H dφΓ =∫

RΓ

F∑

x∈Γ

∂

∂φ(x)

{∂G

∂φ(x)e−H

}dφΓ

= −∑

x∈Γ

∫

RΓ

∂F

∂φ(x)

∂G

∂φ(x)· e−H dφΓ ,

for every F = F (φ), G = G(φ) ∈ C2b (RΓ ). The Hamiltonians H may be moregeneral than (2.1), for instance, those with self potentials, see (6.3)

Remark 2.3 (1) The dynamics corresponding to the massive HamiltonianHm (recall Remark 2.2) can be introduced similarly. It forces the heightsφ = {φ(x)} to stay bounded.(2) Interface dynamics of SOS type was studied by several authors, e.g., Dun-lop [90] considered the dynamics for the corresponding gradient fields in onedimension; see also Remark 13.1 and Sect. 16.5.

2.5 Scaling limits

Our main interest is in the analysis of the scaling limit, which passes frommicroscopic to macroscopic levels. For the microscopic height variables φ ={φ(x);x ∈ Γ} with Γ = DN ,TdN or Zd, the macroscopic height variableshN = {hN(θ)} are associated by

hN (θ) =1

Nφ ([Nθ]) , θ ∈ D,Td or Rd, (2.16)

where [Nθ] stands for the integer part of Nθ(∈ Rd) taken componentwise.Note that both x- and φ-axes are rescaled by a factor 1/N . This is becausethe ϕ-field represents a hypersurface embedded in d + 1 dimensional space.The functions hN are step functions. Sometimes interpolations by polilinearfunctions (or polygonal approximations) are also considered, see (6.9) and(6.21) below.

For the time evolution φt = {φt(x);x ∈ Γ}, t > 0 of the interface, we shallmostly work under the space-time diffusive scaling

hN (t, θ) =1

NφN2t([Nθ]). (2.17)

2.6 Quadratic potentials

Here we take a quadratic function V (η) = 12η2 as a typical example of the

potential satisfying our basic conditions (2.2). To rewrite the HamiltonianH(φ) for such V , let us introduce the discrete Laplacian ∆ ≡ ∆Λ,ψ for Λ b Zdwith boundary conditions ψ ∈ R∂+Λ

∆φ(x) =∑

y∈Λ:|x−y|=1

((φ ∨ ψ)(y) − φ(x)) , x ∈ Λ, (2.18)


where φ ∨ ψ ∈ RΛ stands for the height variables which coincide with φ on Λand with ψ on ∂+Λ, respectively; i.e., φ ∨ ψ(x) = φ(x) for x ∈ Λ and = ψ(x)for x ∈ ∂+Λ. The summation by parts formula proves that

H0Λ(φ) = −1

2(φ,∆Λ,0φ)Λ (2.19)

where (φ1, φ2)Λ =∑x∈Λ φ1(x)φ2(x) denotes an inner product of φ1 and φ2 ∈

RΛ. The boundary condition is taken ψ = 0 for simplicity. In particular, the

finite volume Gibbs measure µ0Λ can be expressed as

µ0Λ(dφΛ) =1

Z0Λe

12 (φ,∆Λ,0φ)Λ dφΛ,

and accordingly, φΛ forms a Gaussian field under the distribution µ0Λ with

mean 0 and covariance (−∆Λ,0)−1, the inverse operator of −∆Λ,0, see Sect.3.1 for more details.

For V (η) = 12η2, the corresponding dynamics (2.9) is a simple discrete

stochastic heat equation

dφt(x) = ∆φt(x)dt+√

2dwt(x), x ∈ Γ. (2.20)

22 T. Funaki

3 Gaussian equilibrium systems

As a warming up before studying general systems, let us consider the ∇ϕinterface model in the case where the potential is quadratic: V (η) = 12η

2. Thecorresponding system formed by the height variables φ is then Gaussian andsometimes called a free lattice field or a harmonic oscillator in physical litera-tures. For a Gaussian system, one can explicitly compute the mean, covariance(two-point correlation function) and characteristic functions. In particular, aswe shall see, the covariance of our field φ can be represented by means of thesimple random walks on the lattice, Proposition 3.2. This will be extended togeneral potentials V and called the Helffer-Sjöstrand representation, see Sect.4.1 below.

We begin with systems on finite and connected regions Λ(b Zd) in Sect. 3.1and then, by taking the thermodynamic limit (i.e., Λ↗ Zd), infinite systemson Zd will be constructed in Sect. 3.2. We shall also discuss massive systemand see significant differences in massive and massless systems, for instance,in the speed of decay of correlation functions or the dependence of the systemon the boundary conditions, see Sect. 3.3. Sect. 3.4 deals with the macroscopicscaling limits for ϕ and ∇ϕ-fields.

3.1 Gaussian systems in a finite region

We assume that Λ b Zd is connected. When V (η) = 12η2 and the boundary

conditions ψ ∈ R∂+Λ (or ψ ∈ RZd or ψ ∈ RΛc) are given, the correspondingHamiltonian H(φ) ≡ HψΛ (φ) defined by (2.1) is a quadratic form of φ so thatthe finite volume ϕ-Gibbs measure µψΛ ∈ P(RΛ) (or ∈ P(RZ

d

)) determined by(2.4) is Gaussian.

Harmonic functions and Green functions

The mean and covariance of the height variables φ = {φ(x);x ∈ Λ} under µψΛare computable by solving the Dirichlet boundary value problem on Λ for thediscrete Laplacian ∆. Indeed, we consider the difference equation on Λ withthe boundary condition ψ

∆φ(x) :=∑

y∈Zd:|x−y|=1(φ(y) − φ(x)) = 0, x ∈ Λ,

φ(x) =ψ(x), x ∈ ∂+Λ,(3.1)

which is equivalent to

∆Λ,ψφ(x) = 0, x ∈ Λ,

where ∆Λ,ψ is the discrete Laplacian determined by (2.18). The solution φ ≡φΛ,ψ = {φ(x);x ∈ Λ} of (3.1) is unique and called a (discrete) harmonicfunction on Λ.


Let GΛ(x, y), x ∈ Λ, y ∈ Λ be the Green function (potential kernel) forthe discrete Laplacian ∆Λ,0 with boundary condition 0, i.e., the solution ofequations

{−∆GΛ(x, y) = δ(x, y), x ∈ Λ,

GΛ(x, y) = 0, x ∈ ∂+Λ,(3.2)

where δ(x, y) is the Kronecker’s δ, and ∆ acts on the variable x and y isthought of as a parameter. In fact, {GΛ(x, y);x, y ∈ Λ} is the inverse matrixof {−∆Λ(x, y);x, y ∈ Λ} so that we shall denote

GΛ(x, y) = (−∆Λ)−1(x, y),

note that ∆Λ(x, y) is the kernel of ∆Λ ≡ ∆Λ,0: ∆Λφ(x) =∑

y∈Λ∆Λ(x, y)φ(y).

Mean, covariance and characteristic functions

The next proposition is an extension of the fact stated in Sect. 2.6 when theboundary conditions are ψ ≡ 0.Proposition 3.1 (1) Under µψΛ, φ = {φ(x);x ∈ Λ} is Gaussian with meanφΛ,ψ = {φΛ,ψ(x);x ∈ Λ} and covariance GΛ(x, y), i.e., µψΛ = N(φΛ,ψ, GΛ).In particular, for x, y ∈ Λ

EµψΛ [φ(x)] = φΛ,ψ(x), (3.3)

EµψΛ [φ(x);φ(y)] = GΛ(x, y), (3.4)

where

Eµ [φ(x);φ(y)] := Eµ [{φ(x) −Eµ[φ(x)]} {φ(y) −Eµ[φ(y)]}]

stands for the covariance of φ(x) and φ(y) under µ.

(2) The characteristic function of µψΛ is given by

EµψΛ

[e√−1(ξ,φ)Λ

]= exp

{√−1(ξ, φΛ,ψ)Λ −

1

2(ξ, (−∆Λ)−1ξ)Λ

}

for ξ ∈ RΛ.(3) If φ is µ0Λ-distributed, then φ+ φΛ,ψ is µ

ψΛ-distributed.

Proof. A careful rearrangement of the sum in the HamiltonianHψΛ (φ) applyingthe summation by parts formula leads us to

HψΛ (φ) = −1

2

((φ− φΛ,ψ), ∆Λ(φ− φΛ,ψ)

)Λ

+1

2

∑

x∈Λ,y/∈Λ|x−y|=1

φΛ,ψ(y)∇φΛ,ψ(〈y, x〉),

24 T. Funaki

for every φ ∈ RΛ. This is an extension of (2.19) for 0-boundary conditionsand a discrete analogue of Green-Stokes’ formula. Note that the second termin the right hand side depends only on the boundary conditions ψ and not onφ. Therefore, we have that

µψΛ(dφΛ) =1

Z̃ψΛexp

{1

2((φ − φΛ,ψ), ∆Λ(φ− φΛ,ψ))Λ

}dφΛ

with a proper normalization constant Z̃ψΛ . This immediately shows the asser-tions (1) and (2). The third assertion (3) follows from (1) or (2).

It might be useful to give another proof for (1). Actually, to show (3.3),set its left hand side as h(x). Then, h(x) satisfies the equation (3.1). In fact,the boundary condition is obvious and, for x ∈ Λ,

∆h(x) = EµψΛ [∆φ(x)] = −EµψΛ

[∂HψΛ∂φ(x)

]= 0

by the integration by parts under µψΛ. The uniqueness of solutions of (3.1)proves (3.3). The proof of (3.4) is similar; one may check its left hand sidesolves (3.2) in place of GΛ(x, y). This can be shown again by the integrationby parts.

It is standard to calculate the mean, covariance and other higher momentsfrom the characteristic function. Indeed, for instance, (3.4) has the third proof:We may assume ψ ≡ 0 by translating the field φ by φΛ,ψ and in this case

Eµ0Λ [(ξ, φ)2Λ] = −

d2

dα2Eµ

0Λ

[e√−1α(ξ,φ)Λ

] ∣∣∣∣∣α=0

= − d2

dα2e−

α2

2 (ξ,(−∆Λ)−1ξ)Λ

∣∣∣∣∣α=0

= (ξ, (−∆Λ)−1ξ)Λ.

Then, the identity (3.4) follows by taking ξ = δx, δy or δx+ δy in this formulaand computing their differences, where δx(·) = δ(x, ·). ut

In particular, for µN ≡ µ0DN with Λ = DN taking D = (−1, 1)d and with0-boundary conditions, we have EµN [φ(O)] = 0 and the variance behaves asN → ∞

EµN [φ(O)2] = (−∆DN )−1(O,O) ≈

1, d ≥ 3,logN, d = 2,N, d = 1,

(3.5)

where ≈ means that the ratio of the both sides stays uniformly positive andbounded. The number of the sites neighboring to each site is 2d and thereforeone can expect that, as the lattice dimension d increases, the fluctuations ofthe interfaces become smaller, in other words, they gain more stiffness. The


behavior (3.5) of the variance agrees with this observation. When d ≥ 3, thesecond moment stays bounded as N → ∞ and accordingly ϕ-Gibbs measureis normalizable in the sense that it admits the thermodynamic limit, see Sect.3.2. For general convex potentials V , Brascamp-Lieb inequality gives at leastthe corresponding upper bound in (3.5), see Sect. 4.2. When d = 1, φ(x) isessentially the pinned Brownian motion with discrete time parameter x ∈(−N,N) ∩ Z and therefore (3.5) is standard.

Random walk representation

Let X = {Xt}t≥0 be the simple random walk on Zd with continuous timeparameter t, i.e., the generator of X is the discrete Laplacian ∆ and the jumpof X to the adjacent sites is accomplished by choosing one of them with equalprobabilities after an exponentially distributed waiting time with mean 12d .Let τΛ be the exit time of X from the region Λ:

τΛ := inf{t ≥ 0; Xt ∈ Λc}.

The transition probability of the simple random walk on Λ with absorbingboundary ∂+Λ is denoted by pΛ(t, x, y) ≡ Ex[1{y}(Xt), t < τΛ], t ≥ 0, x, y ∈Λ, where Ex[ · ] stands for the expectation for X starting at x: X0 = x. Then,the following representations are easy.

Proposition 3.2 For every x, y ∈ Λ, we have

φΛ,ψ(x) = Ex [ψ (XτΛ)] , (3.6)

GΛ(x, y) = Ex

[∫ τΛ

0

1{y} (Xt) dt

]=

∫ ∞

0

pΛ(t, x, y) dt. (3.7)

The middle term of (3.7) is the average of the occupation time of Xt at ybefore leaving Λ.

3.2 Gaussian systems on Zd

Let us assume that a harmonic function ψ = {ψ(x);x ∈ Zd} ∈ RZd is givenon the whole lattice Zd and consider the Gaussian finite volume ϕ-Gibbs

measures µψΛ ∈ P(RZd

) for all connected Λ b Zd. We shall see that, if d ≥ 3,µψΛ admits a weak limit µ

ψ ∈ P(RZd) as Λ ↗ Zd (i.e., along an increasingsequence {Λ(n)}n=1,2,... satisfying ∪∞n=1Λ(n) = Zd) and the limit µψ is a ϕ-Gibbs measure (on Zd) corresponding to the potential V (η) = 12η

2. A simplebut important class of the harmonic functions on Zd is given by ψ(x) = u·x+hfor u ∈ Rd and h ∈ R, where u · x denotes the inner product in Rd. The two-point correlation function of µψ decays slowly in algebraic (i.e., polynomial)order. The ∇ϕ-Gibbs measures exist for arbitrary dimension d.

26 T. Funaki

Thermodynamic limit

Since φΛ,ψ = ψ on Λ for every harmonic function ψ, from (3.3), the mean of

φ under µψΛ is ψ. The covariance of µψΛ is GΛ(x, y), recall (3.4). Let G(x, y) ≡

(−∆)−1(x, y) = G(x−y) be the Green function (of 0th order) of the operator∆ on Zd, i.e.,

G(x, y) =

∫ ∞

0

p(t, x, y) dt, x, y ∈ Zd,

where p(t, x, y) denotes the transition probability of the simple random walkX on Zd. It is well-known that G(x, y) < ∞ if and only if d ≥ 3 (i.e., if Xis transient). This can be also seen from an explicit formula for G(x) by theFourier transform:

G(x) =1

2(2π)d

∫

T̃d

e√−1x·θ

∑dj=1(1 − cos θj)

dθ, (3.8)

where T̃d = (−π, π]d and dθ = ∏dj=1 dθj . Since pΛ(t, x, y) ↑ p(t, x, y) as Λ ↗Zd, we have

limΛ↗Zd

GΛ(x, y) = G(x, y), x, y ∈ Zd.

To study the limit of µψΛ as Λ↗ Zd, recalling Proposition 3.1-(3), we mayassume ψ ≡ 0. Let µ ∈ P(RZd) be the distribution of a Gaussian systemφ = {φ(x);x ∈ Zd} with mean 0 and covariance G(x, y), whose characteristicfunction is given by

Eµ[e√−1(ξ,φ)

]= e−

12 (ξ,(−∆)

−1ξ), ξ ∈ C0(Zd),

where (ξ, φ) =∑x∈Zd ξ(x)φ(x) is the inner product (on the whole lattice Z

d)

and C0(Zd) denotes the family of all ξ : Zd → R satisfying ξ(x) = 0, x /∈ Λ for

some Λ b Zd. The convergence of the covariances

(ξ, (−∆Λ)−1ξ) =∑

x,y∈ZdGΛ(x, y)ξ(x)ξ(y)

−→∑

x,y∈ZdG(x, y)ξ(x)ξ(y) = (ξ, (−∆)−1ξ)

for ξ ∈ C0(Zd) (note that both sums are finite) implies the convergence of thecharacteristic functions so that µ0Λ weakly converges to µ as Λ ↗ Zd on thespace RZ

d

endowed with the product topology.In fact, the convergence holds under stronger topologies. To see that, let

us introduce weighted `2-spaces on Zd

`2(Zd, z) := {φ ∈ RZd ; ‖φ‖2z :=∑

x∈Zdφ(x)2z(x)


for weight functions z = {z(x) > 0;x ∈ Zd}. We shall especially concernwith two classes of spaces (`2α, ‖ · ‖α) and (`2r, ‖ · ‖r) for α, r > 0 takingz(x) = (1 + |x|)−α and z(x) = e−2r|x|, respectively.Proposition 3.3 Assume d ≥ 3. Then µ0Λ weakly converges to µ as Λ↗ Zdon the spaces `2α, α > d or `

2r, r > 0.

Proof. The proof is concluded once the tightness of {µ0Λ}Λ on these spaces isshown. However, since

0 ≤ Eµ0Λ[φ(x)2

]= GΛ(x, x) ≤ G(x, x) = G(O)

28 T. Funaki

Long correlations

As we have seen, the two-point correlation function of φ under the ϕ-Gibbsmeasure µψ coincides with the Green function G(x, y) of the simple randomwalk on Zd, and it decays only algebraically (or in polynomial order) and notexponentially fast. In this sense the field has long dependence.

Proposition 3.5 Assume d ≥ 3. Then the two-point correlation function ofµψ is always positive and behaves like

Eµψ

[φ(x);φ(y)] ∼ k1|x− y|d−2

as |x− y| → ∞, where |x− y| stands for the Euclidean distance and ∼ meansthat the ratio of both sides converges to 1. The constant k1 is determined by

k1 =1

2

∫ ∞

0

(2πt)−d2 e−

12t dt.

Proof. The conclusion follows from the behavior G(x) ∼ k1/|x|d−2, |x| →∞ of the Green function established by Itô-McKean [155] (2.7, p.121); seealso Spitzer [238], p.308, P1 for d = 3 and Lawler [185]. Note that, in thesereferences, ∆ is normalized by dividing it by 2d. ut

This proposition, in particular, implies that one of the important thermo-dynamic quantities called the compressibility diverges in massless model:

∑

x∈ZdEµ

ψ

[φ(x);φ(y)] = ∞.

Note that k1/|x − y|d−2, x, y ∈ Rd is the Green function (of the continuumLaplacian) on Rd and the constant k1 has another expression

k1 = (4πd/2)−1Γ

(d

2− 1)

=1

(d− 2)Ωd,

where Ωd is the surface area of the d− 1 dimensional unit sphere. For generalpotential V , similar asymptotics for the two-point correlation function areobtained by [202], see Sect. 4.3.

∇ϕ-Gaussian field

We have required the assumption d ≥ 3 to construct ϕ-field on the infinitevolume lattice Zd, but for its gradient the thermodynamic limit exists inarbitrary dimensions d including d = 1, 2. To see this, we first notice the nextlemma which is immediate from Proposition 3.1-(1). Recall that

∇iφ(x) := φ(x + ei) − φ(x)(≡ ∇φ(x + ei)), x ∈ Zd, 1 ≤ i ≤ d.The bond 〈x+ei, x〉 is simply denoted by x+ei and, in particular, ei sometimesrepresents the bond 〈ei, O〉.


Lemma 3.6 Let Λ b Zd and ψ ∈ RZd be given. Then we have

EµψΛ [∇iφ(x)] = ∇iφΛ,ψ(x), (3.9)

EµψΛ [∇iφ(x);∇jφ(y)] = ∇i,x∇j,yGΛ(x, y), (3.10)

for every x, y ∈ Λ, 1 ≤ i, j ≤ d, where ∇i,x and ∇j,y indicate that theseoperators act on the variables x and y, respectively.

When d = 1, 2, although GΛ(x, y) itself is not convergent as Λ ↗ Zd, itsnormalization

G̃Λ(x, y) :=

∫ ∞

0

{pΛ(t, x, y) − pΛ(t, 0, 0)} dt

admits the finite limit

G(x, y) :=

∫ ∞

0

{p(t, x, y) − p(t, 0, 0)} dt, x, y ∈ Zd,

which is called the (normalized 0th order) Green function. One can replaceGΛ in the right hand side of (3.10) with G̃Λ so that the covariance of the∇ϕ-field has the limit as Λ↗ Zd. We therefore obtain the next proposition.

Proposition 3.7 For a harmonic function ψ on Zd, let µψ,∇ ∈ P(R(Zd)∗) bethe distribution of the Gaussian field on (Zd)∗ with mean and covariance

Eµψ,∇

[∇iφ(x)] = ∇iψ(x),Eµ

ψ,∇

[∇iφ(x);∇jφ(y)] = ∇i,x∇j,yG(x, y),

respectively. Then µψ,∇ is a ∇ϕ-Gibbs measure (see Definition 2.2 and Sect.9).

We have a family of ∇ϕ-Gibbs measures {µψu,∇;u ∈ Rd} by takingψ(x) ≡ ψu(x) := u · x. When d ≥ 3, if φ = {φ(x);x ∈ Zd} is µψ-distributed, then its gradient field ∇φ = {∇φ(b); b ∈ (Zd)∗} is µψ,∇-distributed. When d = 1, the Green function is given by G(x) = − 12 |x|, whichproves Eµ

ψ,∇

[∇φ(x);∇φ(y)] = δ(x− y)(≡ δ(x, y)). This, in particular, showsthat {∇φ(b); b ∈ (Z)∗} is an independent Gaussian system in one dimension.When d = 2, the Green function behaves like

G(x) = − 12π

log |x| + c0 +O(|x|−2), |x| → ∞,

see Stöhr [241], Spitzer [238].

30 T. Funaki

3.3 Massive Gaussian systems

In the present subsection, we study the ϕ-field associated with the massiveHamiltonian Hm(φ) introduced in Remark 2.2-(1). The mass term of Hm ac-tually has a strong influence on the field. It is localized and exhibits verydifferent features from the massless case. In particular, (1) the ϕ-Gibbs mea-sure exists for arbitrary dimensions d ≥ 1, (2) the effect of the boundaryconditions is weak (see Corollary 3.9 below) and (3) the two-point correla-tion function decays exponentially fast; in other words, the field has a strongmixing property.

Massive Gaussian ϕ-Gibbs measures

For Λ b Zd and the boundary condition ψ ∈ RZd , the finite volume ϕ-Gibbsmeasure µψΛ;m ∈ P(RΛ) having mass m > 0 is defined by

µψΛ;m(dφΛ) :=1

ZψΛ;me−H

ψΛ;m(φ) dφΛ

where

HψΛ;m(φ) = HψΛ (φ) +

m2

2

∑

x∈Λφ(x)2

is the massive Hamiltonian and ZψΛ;m is the normalization constant. As before,

we sometimes regard µψΛ;m ∈ P(RZd

). The ϕ-Gibbs measure µ ≡ µm ∈ P(RZd

)

(on Zd) having massm is defined by means of the DLR equation with the local

specifications µψΛ;m in place of µψΛ in Definition 2.1. We are always concerning

the case where V (η) = 12η2 throughout this section.

Finite systems

Similarly to the massless case, the mean and covariance of the field φ underµψΛ;m can be expressed as solutions of certain difference equations and admitthe random walk representation. Indeed, consider the equations (3.1) and (3.2)with ∆ replaced by ∆−m2, respectively, i.e.,

{(∆−m2)φ(x) = 0, x ∈ Λ,

φ(x) = ψ(x), x ∈ ∂+Λ,(3.11)

and {−(∆−m2)GΛ;m(x, y) = δ(x, y), x ∈ Λ,

GΛ;m(x, y) = 0, x ∈ ∂+Λ,(3.12)

for y ∈ Λ. The solution of (3.11) is denoted by φ = φΛ,ψ;m, while GΛ;m(x, y)is sometimes written as


GΛ;m(x, y) = (−∆Λ +m2)−1(x, y).

Consider the simple random walkX = {Xt}t≥0 on Zd as before and let σ be anexponentially distributed random variable with mean 1m2 being independentof X . The random walk X is killed at the time σ, in other words, it jumps toa point ∆(/∈ Zd) at σ and stays there afterward. Every function ψ on Zd isextended to Zd ∪ {∆} setting ψ(∆) = 0. The next proposition is an extensionof Propositions 3.1 and 3.2 to the massive case. The proof is similar.

Proposition 3.8 Under µψΛ;m, φ = {φ(x);x ∈ Λ} is Gaussian with meanφΛ,ψ;m(x) and covariance GΛ;m(x, y). In particular, we have for x, y ∈ Λ

EµψΛ;m [φ(x)] = φΛ,ψ;m(x) = Ex [ψ (XτΛ∧σ)] , (3.13)

EµψΛ;m [φ(x);φ(y)] = GΛ;m(x, y) = Ex

[∫ τΛ∧σ

0

1{y} (Xt) dt

]. (3.14)

Thermodynamic limit

The random walk representation is useful to observe that the limit of µψΛ;mas Λ↗ Zd does not depend on the boundary condition ψ if it grows at mostin polynomial order as |x| → ∞. This property for massive field is essentiallydifferent from the massless case. If ψ grows exponentially fast, its effect mayremain in the limit of µψΛ;m, see Remark 3.3 below.

Corollary 3.9 If the function ψ on Zd satisfies |ψ(x)| ≤ C(1+|x|n) for someC, n > 0, then we have for every x ∈ Zd

limΛ↗Zd

φΛ,ψ;m(x) = 0.

Proof. To prove the conclusion, from (3.13), it suffices to show that Px(τΛ` <σ) = o(`−n) as ` → ∞, where Λ` = [−`, `]d ∩ Zd. However, since P (σ >√`) = e−m

2√`, this follows from the large deviation type estimate on τΛ:

Px(τΛ` <√`) ≤ e−C` for some C > 0. ut

The covariance GΛ;m(x, y) of µψΛ;m converges as Λ ↗ Zd to Gm(x, y) =

Gm(x − y), where Gm(x) is defined by

Gm(x) :=

∫ ∞

0

e−m2tp(t, x) dt

=1

(2π)d

∫

T̃d

e√−1x·θ

2∑dj=1(1 − cos θj) +m2

dθ,

where p(t, x) = p(t, x, O). Note that, since m > 0, Gm(x) < ∞ for all d ≥ 1.The function Gm(x, y) is sometimes written as (−∆+m2)−1(x, y) and called

32 T. Funaki

the Green function ofm2th order of the operator∆ on Zd. When the boundarycondition ψ satisfies the condition in Corollary 3.9, the thermodynamic limit

µm ∈ P(RZd

) of µψΛ;m exists and it is the Gaussian measure with the mean 0,covariance Gm(x, y) and characteristic function

Eµm[e√−1(ξ,φ)

]= e−

12 (ξ,(−∆+m

2)−1ξ),

for ξ ∈ C0(Zd). The limit measure is independent of the choice of ψ as longas it satisfies the condition in Corollary 3.9.

Remark 3.3 Benfatto et al. [15] characterized the structure of the class ofall massive Gaussian ϕ-Gibbs measures on RZ when d = 1. Their result showsthat its extremal set E is given by

E = {µα−,α+ ; (α−, α+) ∈ R2},

where µα−,α+ is the limit of the sequence of finite ϕ-Gibbs measures µψ[−`,`];m

with boundary condition ψ satisfying

α± = (1 − ρ2) lim`→∞

ρ`ψ(±(`+ 1))

for certain ρ ∈ (0, 1). For instance, if ψ is replaced by ψ+u·x+h, the constantsα± are the same. In this respect, ϕ-Gibbs measure is not much sensitive to

the boundary conditions. The mass term m2

2

∑φ(x)2 has an effect to localize

the field. In fact, the above mentioned result implies that the shift invariantmassive Gaussian ϕ-Gibbs measure is unique in one dimension.

Short correlations

The exponential decay of the two-point correlation function

Eµm [φ(x);φ(y)] = Eµm [φ(x)φ(y)] = Gm(x− y)

under µm is precisely stated in the next proposition. The proof of (2) is givenby a simple calculation based on the residue theorem.

Proposition 3.10 (1) When d ≥ 2, for each ω ∈ Sd−1 (i.e., |ω| = 1), deter-mine b(ω) = bm(ω) ∈ Rd and γ ∈ R \ {0} by

1

2d

∑

|y|=1eb·y =

m2

2d+ 1,

1

2d

∑

|y|=1yeb·y = γω.

Then, we have


Gm(x) ∼ Cd|x|−d−12 e−bm(x/|x|)·x

as |x| → ∞ for some Cd > 0.(2) When d = 1, Gm(x) has an explicit formula:

Gm(x) =1

2π

∫ π

−π

e√−1xθ

2(1− cos θ) +m2 dθ =e−m̃|x|

2 sinh m̃, x ∈ Z

where m̃ > 0 is the solution of an algebraic equation cosh m̃ = m2

2 + 1. Inparticular, m̃ behaves such that m̃ = m+O(m2) as m ↓ 0.Remark 3.4 Let Cm(x − y) = Cm(x, y), x, y ∈ Rd be the Green function ofm2th order for the (continuum) Laplacian on Rd, i.e.,

Cm(x) := (−∆+m2)−1δ(x) =1

(2π)d

∫

Rd

e√−1x·p

p2 +m2dp, x ∈ Rd.

The function Cm(x) has an expression by means of the modified Bessel func-tions. For example, when d = 3, we have

Cm(x) =1

4π|x|e−m|x|,

and, for general d ≥ 1, it behaves

Cm(x) ∼ const md−32 |x|− d−12 e−m|x|,

as m|x| → ∞, see [138] p.126. Note that the exponential decay rates for Gmand Cm are different, see also [234] p.257 for d = 2.

Proposition 3.10 gives the exact exponential decay rates of the Green func-tion Gm for m > 0. However, in order just to see the exponentially decayingproperty of Gm, one can apply the Aronson’s type estimate on the tran-sition probability p(t, x, y) = p(t, x− y) of the simple random walk on Zd:

p(t, x) ≤ min{C

td/2e−|x|

2/Ct, 1

}, t > 0, x ∈ Zd, (3.15)

for some C > 0; see [202] §2, [50] for general random walks. In fact, we dividethe integral

Gm(x) =

∫ ∞

0

e−m2tp(t, x) dt

into the sum of those on two intervals [0, |x|) and [|x|,∞). Then, on the firstinterval, if x 6= 0, we estimate the integrand as

e−m2tp(t, x) ≤ C

td/2e−|x|

2/2Cte−|x|2/2Ct

≤ Ctd/2

e−1/2Cte−|x|2/2C|x| ≤ const e−|x|/2C,

34 T. Funaki

while on the second

e−m2tp(t, x) ≤ e−m

2t2 e−

m2

2 |x|.

This proves that0 < Gm(x) ≤ Ce−c|x|, x ∈ Zd,

for some c, C > 0. See [202] Theorem B for non-Gaussian case.The Aronson’s type estimate is applicable to the massless case as well and,

though it is weaker than Proposition 3.5, we have the following:

0 < G(x) ≤ C|x|d−2 ,

for some C > 0 when d ≥ 3. In fact, the change of the variables t = |x|2s inthe integral implies

G(x) =

∫ ∞

0

p(t, x) dt ≤∫ ∞

0

C

td/2e−|x|

2/Ct dt =C

|x|d−2∫ ∞

0

1

sd/2e−1/Cs ds.

Note that the last integral converges when d ≥ 3. See [202] Theorem C orTheorem 4.13 in Sect. 4.3 for non-Gaussian case.

3.4 Macroscopic scaling limits

The random field φ = {φ(x);x ∈ Zd} is a microscopic object and our goal isto study its macroscopic behavior. In this subsection, we discuss such problemunder the Gaussian measures µ =: µ0 (massless case, d ≥ 3) and µm,m > 0(massive case, d ≥ 1). Recall that µ and µm are ϕ-Gibbs measures on Zdobtained by the thermodynamic limit with boundary conditions ψ ≡ 0; seeSects. 3.2 and 3.3, respectively.

Scaling limits

Let N be the ratio of typical lengths at macroscopic and microscopic levels.Then the point θ = (θi)

di=1 ∈ Rd at macroscopic level corresponds to the

lattice point [Nθ] := ([Nθi])di=1 ∈ Zd at microscopic level, recall Sect. 2.5.

If x ∈ Zd is close to [Nθ] in such sense that |x − [Nθ]| � N , then x alsomacroscopically corresponds to θ. This means that observing the random fieldφ at macroscopic point θ is equivalent to taking its sample mean around themicroscopic point [Nθ]. Such averaging yields a cancellation in the fluctuationsof φ.

Motivated by these observations, let us consider the sample mean of φover the microscopic region ΛN = (−N,N ]d ∩ Zd, which corresponds to themacroscopic box D = (−1, 1]d:

φN

:=1

(2N)d

∑

x∈ΛNφ(x),

note that (2N)d = |ΛN |. The field φ is distributed under µm for m ≥ 0.


Lemma 3.11 As N → ∞, φN converges to 0 in L2 under µm for all m ≥ 0.

Proof. If we denote G(x) by G0(x), we have for all m ≥ 0

Eµm[(φN)2]

=1

(2N)2d

∑

x,y∈ΛNEµm [φ(x)φ(y)] =

1

(2N)2d

∑

x,y∈ΛNGm(x− y).

However, the Green functions admit bounds for some C, c > 0

0 < Gm(x) ≤

C

|x|d−2 , m = 0, d ≥ 3,

Ce−c|x|, m > 0, d ≥ 1,

which prove the conclusion. ut

This lemma is the law of large numbers for ϕ-field and the next natural

question is to study the fluctuation of φN

around its limit 0 under a properrescaling. As we shall see, the necessary scalings will change according asm = 0 (i.e., massless case) or m > 0 (i.e., massive case) due to the differencein the mixing property of the field φ.

Fluctuations in massive ϕ-Gaussian field

First, let us consider the massive case: m > 0. Then the right scaling for the

fluctuation of φN

will be

Φ̃N := (2N)d2 φ

N ≡ 1(2N)

d2

∑

x∈ΛNφ(x). (3.16)

Since (2N)d2 = |ΛN | 12 , this is the usual scaling for the central limit theorem;

recall that φ = {φ(x);x ∈ Zd} distributed under µm has a “nice” exponentialmixing property when m > 0.

Proposition 3.12 The fluctuation Φ̃N weakly converges to the Gaussian dis-tribution N(0,m−2) with mean 0 and variance m−2 as N → ∞.

Proof. Since Φ̃N is Gaussian distributed with mean 0, the conclusion followsfrom the convergence of its variance:

Eµm[(Φ̃N)2]

=1

(2N)d

∑

x,y∈ΛNGm(x − y) −→

N→∞1

m2,

note (1) in the next remark. ut

36 T. Funaki

Remark 3.5 From Bricmont et al. [39] Proposition A1 (p. 294), we have forµm,m > 0

(1)∑x∈Zd

Eµm [φ(O)φ(x)] = m−2,

(2)∑x∈Zd

Eµm [φ(O)∇iφ(x)] ∼ constm−1 (m ↓ 0),

(3)∑x∈Zd

Eµm [∇iφ(O)∇iφ(x)] is absolutely converging for each m and stays

bounded as m ↓ 0 (see Lemma 3.13 below).However, if i 6= j,(4)

∑x∈Zd

Eµm [∇iφ(O)∇jφ(x)] ∼ const | logm| (m ↓ 0).

Loosely speaking, as m ↓ 0, φ is expected to converge to the massless field sothat its covariances (or those of its gradients) might behave like |x− y|2−d (ormaking its gradients in x), and this may prove that

∑

x∈ZdEµm [∇iφ(O)∇jφ(x)] ≈

∫ Rr(2−d)−2 · rd−1 dr ≈ logR,

where R ≈ m−1 is the correlation length.

Fluctuations in massless ϕ-Gaussian field

Next, let us consider the massless case: m = 0 and d ≥ 3. Let φ be µ0-distributed. Since the variance m−2 of the limit distribution of Φ̃N under µmdiverges as m ↓ 0, the scaling (3.16) must not be correct in the masslesscase. However, if we further scale-down the value of Φ̃N dividing it by N andintroduce

ΦN :=1

NΦ̃N ≡ 1

(2N)d2 ·N

∑

x∈ΛNφ(x), (3.17)

then it has the limit under µ0. In fact, the variance of ΦN behaves

Eµ0[(ΦN)2]

= 2−dN−d−2∑

x,y∈ΛNG(x− y)

∼ k1N−2∑

x∈ΛN|x|2−d ∼ k1N−2

∫

|θ|≤N|θ|2−d dθ

= k1N−2∫ N

0

r(2−d)+(d−1) dr = O(1).

Therefore, (3.17) is the right scaling when m = 0. This actually coincideswith the interpretation stated in Sect. 2.5: φ = {φ(x);x ∈ Zd} represents theheight of an interface embedded in d + 1 dimensional space so that both x-and φ-axes should be rescaled by the factor 1/N at the same time.

If we introduce random signed measures on Rd by


ΦN (dθ) :=1

Nd2 +1

∑

x∈Zdφ(x)δx/N (dθ), θ ∈ Rd, (3.18)

then ΦN in (3.17) is represented as ΦN = 2−d/2〈ΦN (·), 1D〉, where 〈ΦN (·), f〉stands for the integral of f = f(θ) under the measure ΦN (·). In this way,studying the limit of ΦN is reduced to investigating more general problem forthe properly scaled empirical measures of φ.

Fluctuations in massless ∇ϕ-Gaussian field

When f = f(θ) has the form f = − ∂g∂θi with certain g = g(θ), we can rewrite〈ΦN (·), f〉 as

〈ΦN (·), f〉 =〈ΦN (·),− ∂g

∂θi

〉= −N− d2−1

∑

x∈Zdφ(x)

∂g

∂θi(x/N)

∼ −N−d2−1∑

x∈Zdφ(x) ·N{g((x+ ei)/N) − g(x/N)}

= −N−d2∑

x∈Zd(φ(x − ei) − φ(x))g(x/N)

= N−d2

∑

x∈Zd∇iφ(x)g((x + ei)/N).

The second line is the approximation of ∂g∂θi by its discrete derivatives. This

rearrangement, in particular, implies that the scaling in ΦN (dθ) coincides withthe usual one of the central limit theorem, if one deals with the correspondinggradient fields ∇φ = {∇φ(x);x ∈ Zd} instead of φ.

Thus it is natural to introduce the scaled empirical measures of ∇φ ={∇iφ; 1 ≤ i ≤ d}:

ΨNi (dθ) ≡ ΨNi (dθ;u) :=1

Nd2

∑

x∈Zd{∇iφ(x) − ui}δx/N(dθ). (3.19)

The field ∇φ is µ∇u -distributed, where µ∇u , u = (ui)di=1 ∈ Rd is the ∇ϕ-Gibbsmeasure µψu,∇ having boundary conditions ψ(x) = ψu(x) ≡ u · x obtainedin Proposition 3.7. Note that ui = E

µ∇u [∇iφ(x)] and, since ∇iφ(x) − ui =∇i(φ−ψu)(x), the distribution of ΨNi (dθ;u) under µ∇u coincides with that ofΨNi (dθ; 0) under µ

∇0 . We may therefore assume u = 0 to study the limit. The

limit of the variance Eµ∇0

[〈ΨNi , g〉2

]as N → ∞ can be computed based on

the next lemma.

Lemma 3.13 ∑

y∈ZdEµ

∇0 [∇iφ(O)∇iφ(y)] =

1

d.

38 T. Funaki

Proof. Each term in the sum can be rewritten as

Eµ [(φ(ei) − φ(O))(φ(y + ei) − φ(y))]

=1

(2π)d

∫

T̃d

2e√−1y·θ − e

√−1(y−ei)·θ − e√−1(y+ei)·θ

2∑dj=1(1 − cos θj)

dθ

=1

(2π)d

∫

T̃d

e√−1y·θ 1 − cos θi∑d

j=1(1 − cos θj)dθ,

which implies

d∑

i=1

Eµ∇0 [∇iφ(O)∇iφ(y)] =

1

(2π)d

∫

T̃d

e√−1y·θ dθ = δ(y).

The conclusion is shown by taking the sum in y ∈ Zd of the both sides of thisidentity. ut


4 Random walk representation and fundamental

inequalities

We are now at the position to enter into the study of the ∇ϕ interface modelfor general convex potentials V satisfying the three basic conditions (V1)-(V3)in (2.2). We shall first establish in this section three fundamental tools for ana-lyzing the model, i.e., Helffer-Sjöstrand representation, FKG (Fortuin-Kasteleyn-Ginibre) inequality and Brascamp-Lieb inequality. Helffer-Sjöstrand representation describes for the correlation functions under theGibbs measures by means of a certain random walk in random environments.Its original idea comes from [144], [237]. This representation readily impliesFKG and Brascamp-Lieb inequalities. The latter is an inequality between thevariances of non-Gaussian fields and those of Gaussian fields, which we canexplicitly compute as we have seen in Sects. 3.1 and 3.2. In particular, uniformmoment estimates on the non-Gaussian fields are obtained and these make uspossible to construct ∇ϕ-Gibbs measures on (Zd)∗ (for every d ≥ 1) and ϕ-Gibbs measures on Zd (for d ≥ 3) by passing to the thermodynamic limit.The arguments in this section heavily rely on the convexity of the potentialV , i.e., the attractiveness of the interaction.

4.1 Helffer-Sjöstrand representation and FKG inequality

Idea behind the representation

Let us shortly explain the idea behind the Helffer-Sjöstrand representation.It gives the following identity for the covariance of F = F (φ) and G = G(φ)under the Gibbs measure µ:

Eµ[F ;G] =∑

x∈Zd

∫ ∞

0

Eµ[∂F (x, φ0)∂G(Xt, φt)] dt. (4.1)

In the right hand side, φt = {φt(x);x ∈ Zd} is the ϕ-dynamics defined by theSDEs (2.13) with µ-distributed initial data φ0, Xt is the random walk on Z

d

starting at x with (temporary inhomogeneous) generator Qφt defined by

Qφf(x) =∑

y:|x−y|=1V ′′(φ(x) − φ(y)) {f(y) − f(x)} ,

for f : Zd → R. Indeed, assuming Eµ[G] = 0, let H be the solution of thePoisson equation −LH = G, where L is the generator of φt determined by(2.15) with Γ = Zd. Then, from (2.14)

Eµ[F ;G] = Eµ[F (−LH)] =∑

x∈ZdEµ[∂F (x, φ)∂H(x, φ)]. (4.2)

However, a simple computation (cf. (4.7) below) shows

40 T. Funaki

∂(LH)(x, φ) = L∂H(x, φ) + (Q∂H(·, φ)) (x) ≡ {(L+Q)∂H} (x, φ)

and therefore

∂H(x, φ) = (L+Q)−1∂(LH) = E(x,φ)

[∫ ∞

0

∂G(Xt, φt) dt

].

This implies the identity (4.1). The above argument is rather formal and, inparticular, one should replace the measure µ with the finite volume Gibbsmeasure [77] or with the Gibbs measure for ∇ϕ-field [135]. Note that theconvexity condition on V (i.e., V ′′ ≥ 0) is essential for the existence of therandom walk Xt.

Precise formulation

Let the finite region Λ b Zd and the boundary condition ψ ∈ RZd be given. Weshall consider slightly general Hamiltonian having external field (chemicalpotential) ρ = {ρ(x);x ∈ Λ} ∈ RΛ:

Hψ,ρΛ (φ) = HψΛ (φ) − (ρ, φ)Λ (4.3)

and the corresponding finite volume ϕ-Gibbs measure

µψ,ρΛ (dφΛ) =1

Zψ,ρΛe−H

ψ,ρΛ (φ) dφΛ ∈ P(RΛ), (4.4)

where Zψ,ρΛ is the normalization constant. This generalization will be usefulfor the proof of Brascamp-Lieb inequality, cf. Lemma 4.6 and Theorem 4.9.The operator Lψ,ρΛ defined by

Lψ,ρΛ F (φ) :=eHψ,ρΛ (φ)

∑

x∈Λ

∂

∂φ(x)

{e−H

ψ,ρΛ (φ)

∂F

∂φ(x)

}

=∑

x∈Λ

{∂2F

∂φ(x)2− ∂H

ψ,ρΛ

∂φ(x)

∂F

∂φ(x)

}

for F = F (φ) ∈ C2(RΛ) is symmetric under the measure µψ,ρΛ and the associ-ated Dirichlet form is given by

E(F,G) ≡Eψ,ρΛ (F,G) := −Eµψ,ρΛ

[F · Lψ,ρΛ G

]

=Eµψ,ρΛ [(∂F, ∂G)Λ] . (4.5)

Recall that ∂F is defined by

∂xF (φ) ≡ ∂F (x, φ) :=∂F

∂φ(x)


and(∂F, ∂G)Λ ≡ (∂F, ∂G)Λ(φ) :=

∑

x∈Λ∂F (x, φ)∂G(x, φ).

For each φ ∈ RΛ, the operator QφΛ ≡ Qφ,ψΛ,0 is introduced by

QφΛf(x) :=∑

b∈Λ∗:yb=x

V ′′(∇(φ ∨ ψ)(b))∇(f ∨ 0)(b)

=∑

y∈Λ:|x−y|=1

V ′′(φ(x) − (φ ∨ ψ)(y)){(f ∨ 0)(y) − f(x)},

for x ∈ Λ and f = {f(x);x ∈ Λ} ∈ RΛ under the boundary conditions φ(x) =ψ(x) and f(x) = 0 for x ∈ ∂+Λ. In particular, when V (η) = 12 cη2, c > 0,Qφ,ψΛ,0 = c∆Λ,0, which is independent of φ and ψ. We further consider theoperator

L ≡ Lψ,ρΛ := Lψ,ρΛ +QψΛ,0acting on the functions F = F (x, φ) on Λ × RΛ, where QψΛ,0F (x, φ) :=Qφ,ψΛ,0F (x, φ) is the operator acting on functions with two variables. The next

lemma is simple, but explains the reason why the operator Qφ,ψΛ,0 is useful.

Lemma 4.1 For every x ∈ Λ and F = F (φ), we have

[∂x, Lψ,ρΛ ] ≡ ∂xLψ,ρΛ − Lψ,ρΛ ∂x = −

∑

y∈Λ

∂2Hψ,ρΛ∂φ(x)∂φ(y)

∂y, (4.6)

∂Lψ,ρΛ F (x, φ) = L∂F (x, φ). (4.7)

Proof. (4.6) is obvious from the definition of Lψ,ρΛ . (4.7) follows from (4.6) bynoting the symmetry of V ′′ and

∂2Hψ,ρΛ∂φ(x)∂φ(y)

=

∑

z∈Λ:|x−z|=1

V ′′(φ(x) − (φ ∨ ψ)(z)), x = y,

−V ′′(φ(x) − φ(y)) , |x− y| = 1,0 , otherwise,

for x, y ∈ Λ. ut

Let φt ≡ φρt = {φt(x);x ∈ Λ} be the process on RΛ generated by Lψ,ρΛ ,i.e., the solution of the SDEs (2.9) with Γ = Λ and H = Hψ,ρΛ :

dφt(x) = −∑

y∈Λ:|x−y|=1

V ′(φt(x) − φt(y))dt

+ ρ(x)dt+√

2dwt(x), x ∈ Λ,φt(y) = ψ(y), y ∈ ∂+Λ.

(4.8)

42 T. Funaki

Let Xt, t ≥ 0 be the random walk on Λ (or, more precisely, on Λ ∪{∆}) with temporally inhomogeneous generator QφtΛ (and with killing rate∑

y∈∂+Λ:|x−y|=1 V′′(φt(x) − ψ(y)) at x ∈ ∂−Λ). Then, L is the genera-

tor of (Xt, φt). Note that the random walk Xt exists since its jump rateV ′′(∇(φt ∨ ψ)(b)) is positive from our assumption (V3).

Theorem 4.2 (Helffer-Sjöstrand representation) The correlation func-

tion of F = F (φ) and G = G(φ) under µψ,ρΛ has the representation

Eµψ,ρΛ [F ;G] =

∑

x∈Λ

∫ ∞

0

Eδx⊗µψ,ρΛ [∂F (x, φ0)∂G(Xt, φt), t < τΛ] dt. (4.9)

In the right hand side, δx ⊗ µψ,ρΛ indicates the initial distribution of (Xt, φt)and δx ∈ P(Λ) is defined by δx(z) = δ(z − x). In particular, the distributionof φ0 is µ

ψ,ρΛ and the random walk Xt starts at x. τΛ = inf{t > 0;Xt ∈ Λc} is

the exit time of Xt from Λ.

Theorem 4.2 with a special choice of F (φ) = φ(x) and G(φ) = φ(y) givesthe following extension of the formula (3.7) combined with (3.4) for quadraticpotentials to general ones; note that ∂F (z, φ) = δ(x− z) in this case.

Corollary 4.3 For every x, y ∈ Λ,

Eµψ,ρΛ [φ(x);φ(y)] = Eδx⊗µ

ψ,ρΛ

[∫ τΛ

0

1{y} (Xt) dt

].

The function F = F (φ) on RΛ is called increasing if it satisfies ∂F =∂F (x, φ) ≥ 0 so that it is nondecreasing under the semi-order on RΛ de-termined by “φ1 ≥ φ2, φ1, φ2 ∈ RΛ ⇐⇒ φ1(x) ≥ φ2(x) for every x ∈ Λ”.Theorem 4.2 immediately implies the following inequality.

Corollary 4.4 (FKG inequality) If F and G are both (L2-integrable) in-creasing functions, then we have

Eµψ,ρΛ [F ;G] ≥ 0,

namely,

Eµψ,ρΛ [FG] ≥ Eµψ,ρΛ [F ]Eµψ,ρΛ [G] .

So far, we are concerned with the representation of correlation functions.The next proposition gives the formula for the expectation of φ(x), which isan extension of (3.3) with (3.6) for quadratic potentials. See [77] for the proof.

Proposition 4.5 For x ∈ Λ, we have

EµψΛ [φ(x)] =

∫ 1

0

Eδx⊗µsψΛ [ψ (XτΛ)] ds.


Introducing the external field ρ has an advantage in the next lemma, whichis indeed one of the tricks commonly used in statistical mechanics. We shalldenote the variance of the random variable X under µ by

var (X ;µ) = Eµ[(X −Eµ[X ])2

].

Lemma 4.6 Assume ρ, ν ∈ RΛ. Then we haved

dsEµ

ψ,sρΛ [φ(x)] = Eµ

ψ,sρΛ [φ(x); (ρ, φ)Λ ] , (4.10)

Eµψ,ρΛ

[exp

{(ν, φ)Λ −Eµ

ψ,ρ

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