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

TL;DR

This paper proves that the critical density for phase transition in Activated Random Walks on integer lattices is always less than one, regardless of dimension or sleep rate, with bounds provided for different regimes.

Contribution

It establishes that the critical density is always below one for all dimensions and sleep rates, offering bounds in both small and large sleep rate regimes.

Findings

Critical density is less than 1 in all dimensions.

Upper bounds are provided for small and large sleep rates.

The phase transition behavior is characterized across regimes.

Abstract

Activated Random Walks, on $\mathbb{Z}^d$ for any $d\geqslant 1$, 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 $\lambda$, 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 $\lambda$. We provide upper bounds for the critical density in both the small $\lambda$ and large $\lambda$ regimes.