Large deviations and concentration properties for ∇ϕ interface models

Abstract

We consider the massless field with zero boundary conditions outside D N ≡D∩ (ℤ d /N) (N∈ℤ+), D a suitable subset of ℝ d , i.e. the continuous spin Gibbs measure ℙ N on ℝ ℤd/N with Hamiltonian given by H(ϕ) = ∑ x,y:|x−y|=1 V(ϕ(x) −ϕ(y)) and ϕ(x) = 0 for x∈D N C . 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 ℙ N under the volume constraint, i.e. the constraint that (1/N d ) ∑ x∈DN 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 ? 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 [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.