Random Surfaces: Large Deviations Principles and Gradient Gibbs Measure Classifications

We study (discretized) “random surfaces, ” which are random functions from Z d (or large subsets of Z d) to E, where E is Z or R. Their laws are determined by convex, nearest-neighbor, gradient Gibbs potentials that are invariant under translation by a full-rank sublattice L of Z d; they include many discrete and continuous height function models (e.g., domino tilings, square ice, the harmonic crystal, the Ginzburg-Landau ∇φ interface model, the linear solidon-solid model) as special cases. We prove a variational principle—characterizing gradient phases of a given slope as minimizers of the specific free energy—and an empirical measure large deviations principle (with a unique rate function minimizer) for random surfaces 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 energy gradient phase µu of slope u. Using a new geometric technique called cluster swapping (a variant of the Swendsen-Wang update for Fortuin-Kasteleyn clusters), we show that µu is unique if at least one of the following holds: E = R, d ∈ {1, 2}, there exists a rough gradient phase of slope u, or u

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