The mixing time of the lozenge tiling Glauber dynamics

Abstract

The broad motivation of this work is a rigorous understanding of reversible, local Markov dynamics of interfaces, and in particular their speed of convergence to equilibrium, measured via the mixing time Tmix. In the (d+1)-dimensional setting, d≥2, this is to a large extent mathematically unexplored territory, especially for discrete interfaces. On the other hand, on the basis of a mean-curvature motion heuristics [Hen97, Spo93] and simulations (see [Des02] and the references in [Hen97, Wil04]), one expects convergence to equilibrium to occur on time-scales of order ≈ δ⁻² in any dimension, with δ→0 the lattice mesh.

We study the single-flip Glauber dynamics for lozenge tilings of a finite domain of the plane, viewed as (2+1)-dimensional surfaces. The stationary measure is the uniform measure on admissible tilings. At equilibrium, by the limit shape theorem [CKP01], the height function concentrates as δ→0 around a deterministic profile , the unique minimizer of a surface tension functional. Despite some partial mathematical results [LT15a, LT15b, Wil04], the conjecture Tmix=δ⁻²⁺ᵒ⁽¹⁾ had been proven, so far, only in the situation where φ is an affine function [CMT12]. In this work, we prove the conjecture under the sole assumption that the limit shape φ contains no frozen regions (facets).