Critical density of activated random walks on transitive graphs

Alexandre Stauffer, Lorenzo Taggi

Research output: Contribution to journalArticle

1 Citation (Scopus)

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 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.

LanguageEnglish
Pages2190-2220
Number of pages31
JournalAnnals of Probability
Volume46
Issue number4
Early online date30 Jun 2018
DOIs
StatusPublished - 1 Jul 2018

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Vertex-transitive
Vertex-transitive Graph
Random walk
Simple Random Walk
Graph in graph theory
Strictly
Model
Graph

Keywords

  • Absorbing states phase transitions
  • Interacting particle systems
  • Random walks

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty

Cite this

Critical density of activated random walks on transitive graphs. / Stauffer, Alexandre; Taggi, Lorenzo.

In: Annals of Probability, Vol. 46, No. 4, 01.07.2018, p. 2190-2220.

Research output: Contribution to journalArticle

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