Electronic Communications in Probability

Electronic Communications in Probability > Vol. 15(2010) > Paper 12

On Fixation of Activated Random Walks

Ori Gurel-Gurevich, Microsoft Research
Gideon Amir, University of Toronto

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. In particular, we do not require the particles path distribution to be Markovian. This allows us to answer a question of Rolla and Sidoravicius, in a more general setting then had been previously known (by Shellef).

Full text: PDF

Pages: 119-123

Published on: April 26, 2010

Bibliography

R. Dickman, L.T. Rolla, V. Sidoravicius Activated Random Walkers: Facts, Conjectures and Challenges Journal of Statistical Physics 138 (2010), 126-142. Math. Review number not available.
R. Lyons, Y. Peres Probability on Trees and Networks Cambridge University Press Math. Review number not available.
L.T. Rolla Generalized Hammersley Process and Phase Transition for Activated Random Walk Models Math. Review number not available.
L.T. Rolla, V. Sidoravicius Absorbing-State Phase Transition for Stochastic Sandpiles and Activated Random Walks Math. Review number not available.
E. Shellef, Nonfixation for Activated Random Walks Math. Review number not available.

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Electronic Communications in Probability. ISSN: 1083-589X