Probability distributions related to tilings of non-convex Polygons

TL;DR

This paper investigates the probability distributions of tile fluctuations in non-convex polygonal regions with cuts, using a discrete tacnode kernel to derive new joint distributions in the context of random lozenge tilings.

Contribution

It introduces new probability distributions for tile fluctuations in non-convex polygons with cuts, utilizing the discrete tacnode kernel for the first time in this setting.

Findings

Derived new joint probability distributions for tile fluctuations.

Applied the discrete tacnode kernel to non-convex polygon tilings.

Provided asymptotic analysis of tile behavior in complex regions.

Abstract

This paper is based on the study of random lozenge tilings of non-convex polygonal regions with interacting non-convexities (cuts) and the corresponding asymptotic kernel as in [3] and [4] (discrete tacnode kernel). Here this kernel is used to find the probability distributions and joint probability distributions for the fluctuation of tiles along lines in between the cuts. These distributions are new.