Universality for lozenge tiling local statistics

• 成果类型: Article
• 署名作者: Aggarwal, Amol
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2023.198.3.1
• 发表日期: 2023
• 页码: 881-1012
• 关键词: random domino tilings; bulk universality; plancherel measures; plane partitions; RANDOM MATRICES; schur process; asymptotics; BOUNDARY; lattice; GROWTH

摘要

In this paper we consider uniformly random lozenge tilings of arbitrary domains approximating (after suitable normalization) a closed, simply-connected subset of R-2 with piecewise smooth, simple boundary. We show that the local statistics of this model around any point in the liquid region of its limit shape are given by the infinite-volume, translation-invariant, extremal Gibbs measure of the appropriate slope, thereby confirming a prediction of Cohn-Kenyon-Propp from 2001 in the case of lozenge tilings. Our proofs proceed by locally coupling a uniformly random lozenge tiling with a model of Bernoulli random walks conditioned to never intersect, whose convergence of local statistics has been recently understood by the work of Gorin-Petrov. Central to implementing this procedure is to establish a local law for the random tiling, which states that the associated height function is approximately linear on any mesoscopic scale.