The mixing time of the lozenge tiling Glauber dynamics

Laslier, Benoît; Toninelli, Fabio

doi:10.5802/ahl.181

Laslier, Benoît; Toninelli, Fabio

The mixing time of the lozenge tiling Glauber dynamics

Annales Henri Lebesgue, Volume 6 (2023), pp. 907-940.

Metadata

DOI10.5802/ahl.181

Keywords Mixing time, lozenge tilings, random interfaces, dimer model

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 $T_{m i x}$ . In the $(d + 1)$ -dimensional setting, $d \geq 2$ , 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 $\approx δ^{- 2}$ in any dimension, with $δ \to 0$ 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 $δ \to 0$ around a deterministic profile $ϕ$ , the unique minimizer of a surface tension functional. Despite some partial mathematical results [LT15a, LT15b, Wil04], the conjecture $T_{m i x} = δ^{- 2 + o (1)}$ 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).

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