Random Surfaces

We study the statistical physical properties of (discretized) “random surfaces,” which are random functions from Z (or large subsets of Z) 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; they include many discrete and continuous height function models (e.g., domino tilings, square ice, the harmonic crystal, the GinzburgLandau ∇φ interface model, the linear solid-on-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 SwendsenWang 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 is irrational. When d = 2 and E = Z, we show that the slopes of all smooth phases (a.k.a. crystal facets) lie in the dual lattice of L. In the case E = Z and d = 2, our results resolve and greatly generalize a number of conjectures of Cohn, Elkies, and Propp—one of which is that there is a unique ergodic Gibbs measure on domino tilings for each non-extremal slope. We also prove several theorems cited by Kenyon, Okounkov, and Sheffield in their recent exact solution of the dimer model on general planar lattices. In the case E = R, our results generalize and extend many of the results in the literature on Ginzurg-Landau ∇φ-interface models.