Mixing times for the simple exclusion process with open boundaries

TL;DR

This paper investigates the mixing times of symmetric and asymmetric simple exclusion processes with open boundaries, revealing diverse behaviors including cutoff phenomena and dependence on boundary parameters, using novel multi-species techniques.

Contribution

It provides the first detailed analysis of mixing times for asymmetric exclusion processes with open boundaries, highlighting new behaviors and methodological innovations.

Findings

Mixing times can be linear or exponential depending on boundary rates.

The process exhibits pre-cutoff and cutoff phenomena.

Multi-species particle arguments are introduced as a new technique.

Abstract

We study mixing times of the symmetric and asymmetric simple exclusion process on the segment where particles are allowed to enter and exit at the endpoints. We consider different regimes depending on the entering and exiting rates as well as on the rates in the bulk, and show that the process exhibits pre-cutoff and in some cases cutoff. Our main contribution is to study mixing times for the asymmetric simple exclusion process with open boundaries. We show that the order of the mixing time can be linear or exponential in the size of the segment depending on the choice of the boundary parameters, proving a strikingly different (and richer) behavior for the simple exclusion process with open boundaries than for the process on the closed segment. Our arguments combine coupling, second class particle and censoring techniques with current estimates. A novel idea is the use of multi-species particle arguments, where the particles only obey a partial ordering.