An ensemble related to discrete orthogonal polynomials and its application to tilings of a half-hexagon

TL;DR

This paper explores the asymptotic behavior of discrete probability measures related to particle configurations with a wall, using orthogonal polynomials, and applies these findings to analyze random tilings of a half-hexagon.

Contribution

It introduces a novel connection between discrete orthogonal polynomials and the asymptotic analysis of particle configurations with boundary conditions, with applications to tiling problems.

Findings

Correlations are determinantal and expressed via discrete orthogonal polynomials.

Asymptotic properties of the measures are characterized.

Application to random tilings of a half-hexagon demonstrates the theory's utility.

Abstract

We discuss asymptotic properties of a family of discrete probability measures which may be used to model particle configurations with a wall on a set of discrete nodes. The correlations are shown to be determinantal and are expressed in terms of discrete orthogonal polynomials. As an application we study random tilings of the half-hexagon or, equivalently, configurations of non-intersecting lattice paths above a wall, so called water melons.