Critical density of activated random walks on transitive 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 $\mathbb{Z}^{d}$, $d\geq1$, and that $\mu_{c}<1$ on $\mathbb{Z}$ for small enough sleeping rate. We show that $\mu_{c}\to0$ as $\lambda\to0$ 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.