The Critical Density for Activated Random Walks is always less than 1

Abstract

Activated Random Walks, on ${\mathbb{Z}}^d$ for any $d⩾1$, is an interacting particle system, where particles can be in either of two states: active or frozen. Each active particle performs a continuous-time simple random walk during an exponential time of parameter $\hat{I}»$, after which it stays still in the frozen state, until another active particle shares its location, and turns it instantaneously back into activity. This model is known to have a phase transition, and we show that the critical density, controlling the phase transition, is less than one in any dimension and for any value of the sleep rate $\hat{I}»$. We provide upper bounds for the critical density in both the small $\hat{I}»$ and large $\hat{I}»$ regimes.

44 pages, 1 figure. Version with minor corrections, accepted for publication in Annals of Probability