Exact sampling and fast mixing of Activated Random Walk

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 $ζ$ times the volume of the ball, where $ζ<1$ is the limiting density of the stationary state.

Theorems & Definitions (33)

• Lemma 1
• Lemma 2
• proof
• Lemma 3
• proof
• Theorem 1
• proof
• Lemma 4
• proof
• Lemma 5