### 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 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Statistics and Probability
- Statistics, Probability and Uncertainty

### Cite this

*Annals of Probability*,

*46*(4), 2190-2220. https://doi.org/10.1214/17-AOP1224

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

Research output: Contribution to journal › Article

*Annals of Probability*, vol. 46, no. 4, pp. 2190-2220. https://doi.org/10.1214/17-AOP1224

}

TY - JOUR

T1 - Critical density of activated random walks on transitive graphs

AU - Stauffer, Alexandre

AU - Taggi, Lorenzo

PY - 2018/7/31

Y1 - 2018/7/31

N2 - 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.

AB - 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.

KW - Absorbing states phase transitions

KW - Interacting particle systems

KW - Random walks

UR - http://www.scopus.com/inward/record.url?scp=85048467182&partnerID=8YFLogxK

U2 - 10.1214/17-AOP1224

DO - 10.1214/17-AOP1224

M3 - Article

VL - 46

SP - 2190

EP - 2220

JO - Annals of Probability

JF - Annals of Probability

SN - 0091-1798

IS - 4

ER -