Tilings of non-convex Polygons, skew-Young Tableaux and determinantal Processes

TL;DR

This paper investigates the asymptotic behavior of random lozenge tilings of non-convex polygons, revealing new kernels and statistics arising from the interaction of non-convexities, with potential universality in the limit.

Contribution

It introduces a new analysis of non-convex polygon tilings, deriving explicit integral kernels and exploring their universal limiting behavior.

Findings

Derived explicit multiple integral kernels for finite tilings.

Identified new fluctuation statistics due to non-convex interactions.

Proposed a universal limiting kernel for large-scale non-convex tilings.

Abstract

This paper studies random lozenge tilings of general non-convex polygonal regions. We show that the pairwise interaction of the non-convexities leads asymptotically to new kernels and thus to new statistics for the tiling fluctuations. The precise geometrical figure here consists of a hexagon with cuts along opposite edges. For this model we take limits when the size of the hexagon and the cuts tend to infinity, while keeping certain geometric data fixed in order to guarantee interaction beyond the limit. We show in this paper that the kernel for the finite tiling model can be expressed as a multiple integral, where the number of integrations is related to the fixed geometric data above. The limiting kernel is believed to be a universal master kernel.