Cutoff for activated random walk

TL;DR

This paper establishes a cutoff phenomenon in the mixing time of activated random walk on an interval, occurring at a time proportional to the interval length and the critical density, using a novel graph mixing result.

Contribution

It introduces a new mixing time cutoff result for activated random walk on intervals, linking it to the critical density and activity distribution.

Findings

Mixing time exhibits cutoff at n times the critical density.

A new graph mixing result shows the chain is mixed when activity is widespread.

The proof applies to both uniform and central driving scenarios.

Abstract

We prove that the mixing time of driven-dissipative activated random walk on an interval of length $n$ with uniform or central driving exhibits cutoff at $n$ times the critical density for activated random walk on the integers. The proof uses a new result for arbitrary graphs showing that the chain is mixed once activity is likely at every site.