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.

63 pages, 13 figures; Versions 2, 3: Minor edits and added explanations

References in corpus (2)

• Local limits of lozenge tilings are stable under bounded boundary height perturbations
• Strong Characterization for the Airy Line Ensemble

Cited by in corpus (3)

• Double Interlacing in Random Tiling Models
• KPZ fluctuations in the planar stochastic heat equation
• Tail estimates for the stationary stochastic six vertex model and ASEP