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arXiv · 2108.12874

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

Amol Aggarwal·Jiaoyang Huang
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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.

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Amol Aggarwal, Jiaoyang Huang. 2021-08-29. Edge Statistics for Lozenge Tilings of Polygons, II: Airy Line Ensemble. https://doi.org/10.1017/fmp.2024.16

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