Random surfaces: Large deviations principles and gradient Gibbs measure classifications.

摘要

We study (discretized) “random surfaces,” which are random functions from $Zd$ (or large subsets of $Zd$) to either $Z$ or $R$. Their laws are determined by convex, nearest-neighbor, gradient Gibbs potentials (which may assume the value ∞) that are invariant under translation by a full-rank sublattice $L$ of $Zd$. Our class of models includes several classical discrete statistical physics models with height function representations (e.g., domino tilings, lozenge tilings, and square ice) and several continuous random surface models (e.g., the harmonic crystal, the Ginzburg-Landau ∇&phis; interface model, and the linear solid-on-solid interface model) as special cases.; A gradient phase is an $L$-ergodic, gradient Gibbs measure with finite specific free energy. A gradient phase is smooth if it is the gradient of an ordinary Gibbs measure; otherwise it is rough. For a general class of convex potentials, we prove: (1) A variational principle characterizing gradient phases of a given expected slope as minimizers of the specific free energy. (2) A large deviations principle—with a unique rate function minimizer—for the macroscopic shapes and empirical measure profiles of random surfaces defined on mesh approximations of bounded domains.; We also prove that the surface tension is strictly convex and that if u is in the interior of the space of finite-surface tension slopes, then there exists a minimal gradient phase μ_u of slope u. This μ_u is unique if at least one of the following holds: E = $R$, d ∈ {lcub}1, 2{rcub}, there exists a rough minimal gradient phase of slope u, or one of the d components of u is irrational. Furthermore, when d = 2 and E = $Z$, we show that the slopes of all smooth phases lie in the dual lattice of $L$.