Non-intersecting, simple, symmetric random walks and the extended Hahn kernel

TL;DR

This paper studies non-intersecting symmetric random walks and their connection to determinantal processes, rhombus tilings, and Brownian motions, revealing new links between combinatorics, probability, and random matrix theory.

Contribution

It introduces a determinantal point process for non-intersecting walks with a correlation kernel based on Hahn polynomials and explores their scaling limits to Brownian motions.

Findings

Correlation kernel expressed via Hahn polynomials

Scaling limits relate to Dyson's Hermitian Brownian motion

Connections between tilings, partitions, and random matrices

Abstract

Consider $a$ particles performing simple, symmetric, non-intersecting random walks, starting at points $2(j-1)$, $1\le j\le a$ at time 0 and ending at $2(j-1)+c-b$ at time $b+c$. This can also be interpreted as a random rhombus tiling of an $abc$-hexagon, or as a random boxed planar partition confined to a rectangular box with side lengths $a$, $b$ and $c$. The positions of the particles at all times gives a determinantal point process with a correlation kernel given in terms of the associated Hahn polynomials. In a suitable scaling limit we obtain non-intersecting Brownian motions which can be related to Dysons's Hermitian Brownian motion via a suitable transformation.