Noncolliding system of continuous-time random walks

TL;DR

This paper investigates a noncolliding system of continuous-time symmetric random walks on integers, demonstrating its determinantal structure, expressing correlations via Bessel functions, and exploring its relaxation to equilibrium with the sine kernel.

Contribution

It establishes the determinantal nature of the noncolliding continuous-time random walks and extends the model to infinite particles with a relaxation to equilibrium.

Findings

System is determinantal for finite initial configurations

Correlation kernel expressed with modified Bessel functions

System relaxes to equilibrium with sine kernel

Abstract

The continuous-time random walk is defined as a Poissonization of discrete-time random walk. We study the noncolliding system of continuous-time simple and symmetric random walks on ${\mathbb{Z}}$. We show that the system is determinantal for any finite initial configuration without multiple point. The spatio-temporal correlation kernel is expressed by using the modified Bessel functions. We extend the system to the noncolliding process with an infinite number of particles, when the initial configuration has equidistant spacing of particles, and show a relaxation phenomenon to the equilibrium determinantal point process with the sine kernel.