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

Abstract

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 $n^\delta$ of its limit shape, and the tiling arctic boundary is within $n^{1/3+\delta}$ to its limit shape, for arbitrarily small $\delta>0$. 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.