Edge Statistics for Lozenge Tilings of Polygons, I: Concentration of Height Function on Strip Domains ...

• Main Author: Huang, Jiaoyang
• Format: Report
• Language: unknown
• Published: arXiv 2021
• Subjects: Probability math.PR; FOS Mathematics; 60F05, 60K35, 82B20; Arctic
• Online Access: https://dx.doi.org/10.48550/arxiv.2108.12872; https://arxiv.org/abs/2108.12872

In this paper we study uniformly random lozenge tilings of strip domains. Under the assumption that the limiting arctic boundary has at most one cusp, we prove a nearly optimal concentration estimate for the tiling height functions and arctic boundaries on such domains: with overwhelming probability the tiling height function is within $n^δ$ of its limit shape, and the tiling arctic boundary is within $n^{1/3+δ}$ to its limit shape, for arbitrarily small $δ>0$. This concentration result will be used in [AH21] to prove that the edge statistics of simply-connected polygonal domains, subject to a technical assumption on their limit shape, converge to the Airy line ensemble. ... : 90 pages, 21 figures, this is Part I of a two-paper series ...