The mixing time of the lozenge tiling Glauber dynamics

TL;DR

This paper rigorously analyzes the mixing time of lozenge tiling Glauber dynamics, proving it scales as approximately the inverse square of the lattice mesh size under broad conditions, advancing understanding of interface convergence to equilibrium.

Contribution

It proves the conjectured mixing time scaling for lozenge tilings with non-frozen limit shapes, extending previous results limited to affine profiles.

Findings

Mixing time scales as δ^{-2} for non-frozen limit shapes.

Established the conjecture for a broad class of limit shapes.

Provides mathematical validation of heuristic predictions.

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 $T_{mix}$. In the $(d+1)$-dimensional setting, $d\ge2$, 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 and simulations, one expects convergence to equilibrium to occur on time-scales of order $\approx \delta^{-2}$ in any dimension, with $\delta\to0$ 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, the height function concentrates as $\delta\to0$ around a deterministic profile $\phi$, the unique minimizer of a surface tension functional. Despite some partial mathematical results, the conjecture $T_{mix}=\delta^{-2+o(1)}$ has been proven, so far, only in the situation where $\phi$ is an affine function. In this work, we prove the conjecture under the sole assumption that the limit shape $\phi$ contains no frozen regions (facets).