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

arXiv eprints
 Pub Date:
 August 2021
 arXiv:
 arXiv:2108.12874
 Bibcode:
 2021arXiv210812874A
 Keywords:

 Mathematics  Probability;
 Mathematical Physics;
 Mathematics  Combinatorics
 EPrint:
 56 pages, 12 figures