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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 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.

Original language | English |
---|---|

Pages (from-to) | 2190-2220 |

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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