Edge Statistics for Lozenge Tilings of Polygons, II: Airy Line Ensemble ...

• Main Authors: Aggarwal, Amol, Huang, Jiaoyang
• Format: Report
• Language: unknown
• Published: arXiv 2021
• Subjects: Probability math.PR; Mathematical Physics math-ph; Combinatorics math.CO; FOS Mathematics; FOS Physical sciences; Arctic
• Online Access: https://dx.doi.org/10.48550/arxiv.2108.12874; https://arxiv.org/abs/2108.12874

We consider uniformly random lozenge tilings of simply connected polygons subject to a technical assumption on their limit shape. We show that the edge statistics around any point on the arctic boundary, that is not a cusp or tangency location, converge to the Airy line ensemble. Our proof proceeds by locally comparing these edge statistics with those for a random tiling of a hexagon, which are well understood. To realize this comparison, we require a nearly optimal concentration estimate for the tiling height function, which we establish by exhibiting a certain Markov chain on the set of all tilings that preserves such concentration estimates under its dynamics. ... : 57 pages, 12 figures; Version 2: Minor edits ...