Active phase for activated random walks on $\mathbb{Z}^d$, $ d \geq 3$, with density less than one and arbitrary sleeping rate

TL;DR

This paper proves that the critical density for activated random walks on high-dimensional integer lattices and certain graphs is less than one, confirming a key conjecture and revealing phase transitions on non-amenable graphs.

Contribution

It establishes the conjecture for $ ext{Z}^d$ with $d ext{ } ext{≥} ext{ } 3$ and transitive graphs, and extends phase transition results to non-amenable graphs.

Findings

Critical density is less than one on $ ext{Z}^d$, $d ext{ } ext{≥} ext{ } 3$

Phase transition occurs on non-amenable graphs

Results apply to graphs with transient random walk behavior

Abstract

It has been conjectured that the critical density of the Activated Random Walk model is strictly less than one for any value of the sleeping rate. We prove this conjecture on $\mathbb{Z}^d$ when $d \geq 3$ and, more generally, on graphs where the random walk is transient. Moreover, we establish the occurrence of a phase transition on non-amenable graphs, extending previous results which require that the graph is amenable or a regular tree.