Nonfixation for Activated Random Walks

Understand

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.