Mixing of the exclusion process with small bias

TL;DR

This paper studies how the mixing time of a biased exclusion process on a path changes as the bias diminishes with increasing path length, revealing phase transitions at specific bias scales.

Contribution

It characterizes the mixing time behavior and identifies phase transitions as the bias tends to zero in the biased exclusion process.

Findings

The process exhibits pre-cutoff behavior.

Mixing time transitions occur at bias scales of 1/n and log n/n.

The chain interpolates between unbiased and constant bias regimes.

Abstract

We analyze the mixing behavior of the biased exclusion process on a path of length $n$ as the bias $\beta_n$ tends to $0$ as $n \to \infty$. We show that the sequence of chains has a pre-cutoff, and interpolates between the unbiased exclusion and the process with constant bias. As the bias increases, the mixing time undergoes two phase transitions: one when $\beta_n$ is of order $1/n$, and the other when $\beta_n$ is order $\log n/n$.