Nonfixation for Activated Random Walks

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.

13 pages; Removed Theorem of density=1 as it was valid only for d=1, which was already shown in arXiv:0908.1152v1

References in corpus (1)

• Activated Random Walkers: Facts, Conjectures and Challenges

Cited by in corpus (7)

• Activated Random Walkers: Facts, Conjectures and Challenges
• Non-equilibrium Phase Transitions: Activated Random Walks at Criticality
• Activated Random Walks on ${\mathbb{Z}}^d$
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• Exact sampling and fast mixing of Activated Random Walk
• Universality and Sharpness in Absorbing-State Phase Transitions
• Active Phase for Activated Random Walks on the Lattice in all Dimensions