Critical density of activated random walks on transitive graphs

2015/12/08 by Stauffer, Alexandre, Taggi, Lorenzo · 3 citations
#60K35 #82C22 #82C26 #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR)

paper · doi:10.48550/arxiv.1512.02397

Abstract

We consider the activated random walk model on general vertex-transitive 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 μ_c>0 in any vertex-transitive amenable graph, and that μ_c∈(0,1) for any sleeping rate on regular trees.

Cited by

• Asymptotic behavior of the critical density of activated random walk
• The odometer in subcritical activated random walk
• Mean-field limit for activated random walk on the integer lattice

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