Lozenge tilings, Glauber dynamics and macroscopic shape

• Published in: Communications in Mathematical Physics
• Main Authors: Laslier, Benoit, Toninelli, Fabio Lucio
• Other Authors: Institut Camille Jordan Villeurbanne (ICJ), École Centrale de Lyon (ECL), Université de Lyon-Université de Lyon-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Jean Monnet - Saint-Étienne (UJM)-Centre National de la Recherche Scientifique (CNRS), Probabilités, statistique, physique mathématique (PSPM), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Jean Monnet - Saint-Étienne (UJM)-Centre National de la Recherche Scientifique (CNRS)-École Centrale de Lyon (ECL)
• Format: Article in Journal/Newspaper
• Language: English
• Published: HAL CCSD 2015
• Subjects: [MATH.MATH-PR]Mathematics [math]/Probability [math.PR]; Arctic
• Online Access: https://hal.science/hal-00943611; https://doi.org/10.1007/s00220-015-2396-7

38 pages, 5 figures We study the Glauber dynamics on the set of tilings of a finite domain of the plane with lozenges of side 1/L. Under the invariant measure of the process (the uniform measure over all tilings), it is well known that the random height function associated to the tiling converges in probability, in the scaling limit $L\to\infty$, to a non-trivial macroscopic shape minimizing a certain surface tension functional. According to the boundary conditions the macroscopic shape can be either analytic or contain "frozen regions" (Arctic Circle phenomenon). It is widely conjectured, on the basis of theoretical considerations, partial mathematical results and numerical simulations for similar models, that the Glauber dynamics approaches the equilibrium macroscopic shape in a time of order $L^{2+o(1)}$. In this work we prove this conjecture, under the assumption that the macroscopic equilibrium shape contains no "frozen region".