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Abstract
We consider the activated random walk model on general vertextransitive graphs. A central question in this model is whether the critical density μ _{c} for sustained activity is strictly between 0 and 1. It was known that μ _{c} > 0 on Z ^{d}, d = 1, and that μ _{c} < 1 on Z for small enough sleeping rate. We show that μ _{c} → 0 as λ → 0 in all vertextransitive transient graphs, implying that μ _{c} < 1 for small enough sleeping rate. We also show that μ _{c} < 1 for any sleeping rate in any vertextransitive graph in which simple random walk has positive speed. Furthermore, we prove that μc > 0 in any vertextransitive amenable graph, and that μ _{c} ∞ (0, 1) for any sleeping rate on regular trees.
Original language  English 

Pages (fromto)  21902220 
Number of pages  31 
Journal  Annals of Probability 
Volume  46 
Issue number  4 
Early online date  30 Jun 2018 
DOIs  
Publication status  Published  31 Jul 2018 
Keywords
 Absorbing states phase transitions
 Interacting particle systems
 Random walks
ASJC Scopus subject areas
 Statistics and Probability
 Statistics, Probability and Uncertainty
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Alexandre Stauffer
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