Large deviations and concentration properties for Δφ interface models

We consider the massless field with zero boundary conditions outside D-N = D boolean AND (Z(d)/N) (N is an element of 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(phi) = Sigma(x,y:\x-y\=1) V(phi(x) - phi(y)) and phi(x) = 0 for x is an element of 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: phi 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 xi(N) = phi/N. We study various concentration and relaxation properties of the family of random surfaces {xi(N)} and of the induced family of gradient fields {del(N)xi(N)} as the discretization step 1/N tends to zero (N --> infinity). In particular, we prove a large deviation principle for {xi(N)} and show that the corresponding rate function is given by integral(D) sigma(del u(x))dx, where sigma 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) Sigma(x is an element of DN) xi(N)(x) stays in a neighborhood of a fixed volume upsilon > 0, and the hard-wall constraint, i.e. xi(N)(x) greater than or equal to 0 for all x. This is therefore a model for a droplet of volume upsilon lying above a hard wall. We prove that under these constraints the field {xi(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 {del(N)xi(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 L-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-Sjostrand [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.