Exact sampling and fast mixing of Activated Random Walk

TL;DR

This paper introduces an exact sampling method for Activated Random Walks on finite Euclidean balls, demonstrating that the Markov chain mixes in near-linear time relative to the volume, and conjectures a cutoff phenomenon at a specific density.

Contribution

It provides the first exact sampling algorithm for ARW stationary distribution and establishes near-linear mixing time bounds on Euclidean balls.

Findings

Mixing time is at most 1+o(1) times the volume of the ball.

An exact sampling algorithm for the stationary distribution is developed.

Conjecture of cutoff at a density less than 1 times the volume.

Abstract

Activated Random Walk (ARW) is an interacting particle system on the $d$-dimensional lattice $\mathbb{Z}^d$. On a finite subset $V \subset \mathbb{Z}^d$ it defines a Markov chain on $\{0,1\}^V$. We prove that when $V$ is a Euclidean ball intersected with $\mathbb{Z}^d$, the mixing time of the ARW Markov chain is at most $1+o(1)$ times the volume of the ball. The proof uses an exact sampling algorithm for the stationary distribution, a coupling with internal DLA, and an upper bound on the time when internal DLA fills the entire ball. We conjecture cutoff at time $\zeta$ times the volume of the ball, where $\zeta<1$ is the limiting density of the stationary state.