Critical density of activated random walks on $\mathbb{Z}^d$ and general graphs

Alexandre Stauffer, Lorenzo Taggi

Published 2015-12-08Version 1

We consider the activated random walk model on general vertex-transitive graphs. A central question for 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 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 graph in which simple random walk has positive speed. Furthermore, we prove that $\mu_c >0$ in any amenable graph, and that $\mu_c \in (0,1)$ for any sleeping rate on regular trees.