Critical density of activated random walks on transitive graphs

TL;DR

This paper investigates the critical density for sustained activity in activated random walks on vertex-transitive graphs, showing it approaches zero in transient graphs and remains positive in amenable graphs.

Contribution

It establishes the behavior of the critical density in various classes of graphs, including its limit as sleeping rate tends to zero and conditions for being between zero and one.

Findings

Critical density tends to zero as sleeping rate approaches zero in transient graphs.

Critical density is less than one for small sleeping rates on all vertex-transitive graphs.

Critical density is positive in all vertex-transitive amenable graphs.

Abstract

We consider the activated random walk model on general vertex-transitive graphs. A central question in this model is whether the critical density $\mu_c$ for sustained activity is strictly between 0 and 1. It was known that $\mu_c>0$ on $\mathbb{Z}^d$, $d\geq 1$, and that $\mu_c<1$ on $\mathbb{Z}$ for small enough sleeping rate. We show that $\mu_c\to 0$ as $\lambda\to 0$ in all vertex-transitive transient graphs, implying that $\mu_c<1$ for small enough sleeping rate. We also show that $\mu_c<1$ for any sleeping rate in any vertex-transitive graph in which simple random walk has positive speed. Furthermore, we prove that $\mu_c>0$ in any vertex-transitive amenable graph, and that $\mu_c\in(0,1)$ for any sleeping rate on regular trees.