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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 vertex-transitive 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 vertex-transitive graph in which simple random walk has positive speed. Furthermore, we prove that μc > 0 in any vertex-transitive amenable graph, and that μ c ∞ (0, 1) for any sleeping rate on regular trees.
- Absorbing states phase transitions
- Interacting particle systems
- Random walks
ASJC Scopus subject areas
- Statistics and Probability
- Statistics, Probability and Uncertainty