Large deviations and concentration properties for ∇ φ interface models.

Abstract. We consider the massless field with zero boundary conditions outside D[sub N] Is equivalent to D intersection (Z[sup d]/N) (N is an element of Z[sup +]), D a suitable subset of R[sup d], i.e. the continuous spin Gibbs measure P[sub N] on R[sup Z[sup d]/N] with Hamiltonian given by H(phi) = SIGMA[sub x,y:|x - y| = 1] V(phi(x) - phi(y)) and phi(x) = 0 for x is an element of D[sup C, sub 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: phi represents the height of the interface over the base D[sup N]. Due to the choice of scaling of the base, we scale the height with the same factor by setting xi[sub N] = phi/N. We study various concentration and relaxation properties of the family of random surfaces {xi [sub N]} and of the induced family of gradient fields {*(This character cannot be converted to ASCII) [sup N] xi [sub N]} as the discretization step 1/N tends to zero (N to infinity). In particular, we prove a large deviation principle for {xi [sub N]} and show that the corresponding rate function is given by f [sub p] sigma u (x))dx, where 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[sub N] under the volume constraint, i.e. the constraint that (1/N [sup d]) SIGMA [sub x Is equivalent to D [sub N]] xi [sub N (x) stays in a neighborhood of a fixed volume upsilon Is greater than 0, and the hard-wall constraint, i.e. xi [sub N] (x) Is 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 [sub 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. [ABSTRACT FROM AUTHOR]