Large deviations and concentration properties for ∇ϕ interface models

Abstract

We consider the massless field with zero boundary conditions outside D N ≡ D ∩ ‫ޚ(‬ d /N ) (N ∈ ‫ޚ‬ + ), D a suitable subset of ‫ޒ‬ d , i.e. the continuous spin Gibbs measure ‫ސ‬ N on ‫ޒ‬ ‫ޚ‬ d /N with Hamiltonian given by H (ϕ) = x,y:|x−y|=1 V (ϕ(x)−ϕ(y)) and ϕ(x) = 0 for x ∈ D C N . The interaction V is taken to be strictly convex and with bounded second derivative. This is a standard effective model for a (d + 1)-dimensional interface: ϕ represents the height of the interface over the base D N . Due to the choice of scaling of the base, we scale the height with the same factor by setting ξ N = ϕ/N.We study various concentration and relaxation properties of the family of random surfaces {ξ N } and of the induced family of gradient fields {∇ N ξ N } as the discretization step 1/N tends to zero (N → ∞). In particular, we prove a large deviation principle for {ξ N } and show that the corresponding rate function is given by D σ (∇u(x))dx, where σ is the surface tension of the model. This is a multidimensional version of the sample path large deviation principle. We use this result to study the concentration properties of ‫ސ‬ N under the volume constraint, i.e. the constraint that (1/N d ) x∈D N ξ N (x) stays in a neighborhood of a fixed volume v > 0, and the hard-wall constraint, i.e. ξ N (x) ≥ 0 for all x. This is therefore a model for a droplet of volume v lying above a hard wall. We prove that under these constraints the field {ξ N } of rescaled heights concentrates around the solution of a variational problem involving the surface tension, as it would be predicted by the phenomenological theory of phase boundaries. Our principal result, however, asserts local relaxation properties of the gradient field {∇ N ξ N (·)} to the corresponding extremal Gibbs states. Thus, our approach has little in common with traditional large deviation techniques and is closer in spirit to hydrodynamic limit type of arguments. The proofs have both probabilistic and analytic aspects. Essential analytic tools are ‫ތ‬ p estimates for elliptic equations and the theory of Young measures. On the side of probability tools, a central role is played by the Helffer-Sjöstrand (1991): 60K35, 82B24, 35J15 Key words and phrases: Massless fields -Effective interface models -Large deviations -Random walk in Random environment -Wulff variational problem -Winterbottom construction -Linear and nonlinear elliptic PDEs 50 J.-D. Deuschel et al. Mathematics Subject Classification[31] PDE representation for continuous spin systems which we rewrite in terms of random walk in random environment and by recent results of T. Funaki and H. Spohn [25] on the structure of gradient fields.

Critical Behavior of Massless Free Field at Depinning Transition

“…Now, by part 1 of Theorem 2.4, we have ν a,b Λ (A ∩ B e (0) = ∅) ≥ 1 − 2C |log e| −1 , and by the inverse Brascamp-Lieb inequality [7],…”

“…It would be very interesting to extend the analysis to a more general class of interactions V . As remarked in the introduction, for even, strictly convex, C 2 interactions a representation of the covariance, similar to (2.16), also exists [7]. It was used in particular to establish exponential decay of covariances for this class of interactions [16].…”

Strict Convexity of the Free Energy for a Class of Non-Convex Gradient Models

“…Under the condition (1.4), a large deviation principle for the rescaled profile with rate function given in terms of the integrated surface tension has been derived in [6]. Here also the strict convexity of σ is very important.…”

Finite volume approximation of the effective diffusion matrix: The case of independent bond disorder

“…In view of the random walk representation developed in [4], this conjecture would follow from an approximation result along the lines of Theorem 1.1, but for a different type of environment. We refer to [7] for further details.…”

“…On the mesoscopic level, the integral of the surface tension happens to be precisely the large deviation rate function for the appropriately scaled height field [4], and it has been conjectured in [7] that the Hessian of the surface tension governs the equilibrium fluctuations in the corresponding Funaki-Spohn state. In view of the random walk representation developed in [4], this conjecture would follow from an approximation result along the lines of Theorem 1.1, but for a different type of environment.…”