Edge statistics for lozenge tilings of polygons, I: concentration of height function on strip domains

• Published in: Probability theory and related fields Vol. 188; no. 1-2; pp. 337 - 485
• Main Author: Huang, Jiaoyang
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.02.2024; Springer Nature B.V
• Subjects: Domains; Economics; Estimates; Finance; Insurance; Management; Mathematical and Computational Biology; Mathematical and Computational Physics; Mathematics; Mathematics and Statistics; Operations Research/Decision Theory; Phase transitions; Polygons; Probability Theory and Stochastic Processes; Quantitative Finance; Statistics; Statistics for Business; Strip; Theoretical; Tiling; 60K35; 82B20; 60F05
• ISSN: 0178-8051; 1432-2064
• DOI: 10.1007/s00440-023-01238-0

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 Aggarwal and Huang (Edge statistics for lozenge tilings of polygons, II: Airy line ensemble, 2021. arXiv:2108.12874 ) 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.