Universality of cutoff for independent random walks on the circle conditioned not to intersect

TL;DR

This paper demonstrates a universal cutoff phenomenon for a class of non-intersecting particle processes on a circle, showing independence from specific transition probabilities under certain conditions, with applications to dimer models.

Contribution

It establishes the universality of the cutoff phenomenon for exchangeable particle systems on the circle, extending understanding of mixing times in these Markov processes.

Findings

Cutoff phenomenon occurs independently of transition probabilities.

Asymptotic mixing times are derived under sub-Gaussian assumptions.

Application to dimer models on the hexagonal lattice is provided.

Abstract

In the present paper, we consider a class of Markov processes on the discrete circle which has been introduced by K\"onig, O'Connell and Roch. These processes describe movements of exchangeable interacting particles and are discrete analogues of the unitary Dyson Brownian motion: a random number of particles jump together either to the left or to the right, with trajectories conditioned to never intersect. We provide asymptotic mixing times for stochastic processes in this class as the number of particles goes to infinity, under a sub-Gaussian assumption on the random number of particles moving at each step. As a consequence, we prove that a cutoff phenomenon holds independently of the transition probabilities, subject only to the sub-Gaussian assumption and a minimal aperiodicity hypothesis. Finally, an application to dimer models on the hexagonal lattice is provided.