On Fixation of Activated Random Walks

TL;DR

This paper proves that in the Activated Random Walks model on transitive unimodular graphs, if the system fixates, then all particles do so, establishing an upper bound on the critical density and extending results to more general processes.

Contribution

It introduces a general framework for analyzing fixation in particle systems on unimodular graphs, extending previous results to broader settings.

Findings

All particles fixate if the system fixates.

Critical density is at most 1.

Results apply to more general processes than previously known.

Abstract

We prove that for the Activated Random Walks model on transitive unimodular graphs, if there is fixation, then every particle eventually fixates, almost surely. We deduce that the critical density is at most 1. Our methods apply for much more general processes on unimodular graphs. Roughly put, our result apply whenever the path of each particle has an automorphism invariant distribution and is independent of other particles' paths, and the interaction between particles is automorphism invariant and local. This allows us to answer a question of Rolla and Sidoravicius, in a more general setting then had been previously known (by Shellef).