The mixing time of the lozenge tiling Glauber dynamics

Laslier, Benoît; Toninelli, Fabio

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

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, $d\ge 2$, 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 $\approx {\delta }^{-2}$ in any dimension, with $\delta \to 0$ 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 $\delta \to 0$ around a deterministic profile $\varphi$, the unique minimizer of a surface tension functional. Despite some partial mathematical results [LT15a, LT15b, Wil04], the conjecture $T_{mix}={\delta }^{-2+o\left(1\right)}$ had been proven, so far, only in the situation where $\varphi$ is an affine function [CMT12]. In this work, we prove the conjecture under the sole assumption that the limit shape $\varphi$ contains no frozen regions (facets).

La motivation de ce travail est la compréhension mathématique des dynamiques Markoviennes réversibles d’interfaces aléatoires, et de leur vitesse de convergence vers l’équilibre (temps de mélange $T_{mix}$). En dimension $(d+1)$, $d\ge 2$, il s’agit de questions très ouvertes, en particulier pour des interfaces discrètes mais, sur la base d’arguments heuristiques et de simulations numériques, on conjecture que $T_{mix}$ est d’ordre $\approx {\delta }^{-2}$ en toute dimension, si $\delta \to 0$ est le pas du réseau.

Nous étudions une dynamique de Glauber pour les pavages par losanges d’un domaine du plan, vues comme des surface $(2+1)$-dimensionnelles. La mesure stationnaire est la mesure uniforme sur les pavages admissibles. À l’équilibre, la fonction de hauteur se concentre (pour $\delta \to 0$) autour d’un profil déterministe $\varphi$, l’unique minimiseur d’une fonctionnelle de tension de surface. Malgré certains résultats mathématiques partiels [LT15a, LT15b, Wil04], jusqu’ici la conjecture $T_{mix}={\delta }^{-2+o\left(1\right)}$ n’avait été démontrée que dans le cas où $\varphi$ est une fonction affine [CMT12]. Dans ce travail, nous prouvons la conjecture sous la seule hypothèse que la forme limite $\varphi$ ne contienne pas de régions gelées.

Classification : 60K35, 82C20, 52C20
Keywords: Mixing time, lozenge tilings, random interfaces, dimer model

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