Nonfixation for Activated Random Walks

TL;DR

This paper studies the activated random walk model on general graphs, establishing conditions for nonfixation based on initial occupation density, and analyzing critical densities across dimensions.

Contribution

It introduces a domination of ARW by MSIA, provides deterministic conditions for nonfixation, and determines positive critical densities for infinite sleep rate ARW.

Findings

Initial density > 1 implies nonfixation on bounded degree graphs.

Initial density = 1 implies nonfixation in dimensions < 5.

Critical density for infinite sleep rate ARW is positive in all dimensions.

Abstract

We consider the activated random walk (ARW) model where particles follow the path of a general Markov process on a general graph. We prove ARW dominates a simpler process, multiple source internal aggregation (MSIA), and use this to formulate a deterministic sufficient condition on initial occupations for nonfixation of ARW and similar variants. In particular, on bounded degree graphs, initial occupation density greater than one almost surely implies nonfixation, where independence requirements are weakened to ergodic in the case of Euclidean lattices. We show that for Euclidean lattices of dimension lower than five, initial density of exactly one also implies nonfixation. Finally, we prove the critical density for the infinite sleep rate ARW is positive for all dimensions.