Active Phase for Activated Random Walk on $\mathbb Z^2$

TL;DR

This paper proves that in the symmetric Activated Random Walk model on the two-dimensional integer lattice, the critical density for sustained activity is less than one when the sleep rate is sufficiently low, indicating a phase transition.

Contribution

It establishes that for small sleep rates, the critical density in the model is strictly below one, advancing understanding of phase transitions in activated random walks.

Findings

Critical density is less than one for small sleep rates.

Phase transition occurs at subunit density.

Provides new bounds for the model's critical parameters.

Abstract

We show that for small enough sleep rate, the critical density of the symmetric Activated Random Walk model on $\mathbb Z^2$ is strictly less than one.