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 ${\mathbb{Z}}^d$, $d\ge 1$, 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.

22 pages, 1 Figure. Title changed, improved exposition