Edge Statistics for Lozenge Tilings of Polygons, II: Airy Line Ensemble
arXiv:2108.12874 · doi:10.1017/fmp.2024.16
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.
arXiv comment63 pages, 13 figures; Versions 2, 3: Minor edits and added explanations
References in corpus (15)
- Shape Fluctuations and Random Matrices
- A variational principle for domino tilings
- The shape of a typical boxed plane partition
- Local statistics for random domino tilings of the Aztec diamond
- Asymptotics of uniformly random lozenge tilings of polygons. Gaussian free field
- Stability of the Matrix Dyson Equation and Random Matrices with Correlations
- Airy processes with wanderers and new universality classes
- Uniform convergence to the Airy line ensemble
- Minimizers of Convex Functionals Arising in Random Surfaces
- Bulk universality for random lozenge tilings near straight boundaries and for tensor products
- Local limits of lozenge tilings are stable under bounded boundary height perturbations
- Tilings of non-convex Polygons, skew-Young Tableaux and determinantal Processes
- Lozenge tilings of hexagons with cuts and asymptotic fluctuations: a new universality class
- Probability distributions related to tilings of non-convex Polygons
- Strong Characterization for the Airy Line Ensemble