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**Critical density of activated random walks on transitive graphs.**
*(English)*
Zbl 1397.82038

Summary: We consider the activated random walk model on general vertex-transitive graphs. A central question in this model is whether the critical density \(\mu_{c}\) for sustained activity is strictly between 0 and 1. It was known that \(\mu_{c}>0\) on \(\mathbb{Z}^{d}\), \(d\geq1\), and that \(\mu_{c}<1\) on \(\mathbb{Z}\) for small enough sleeping rate. We show that \(\mu_{c}\rightarrow0\) as \(\lambda\rightarrow0\) in all vertex-transitive transient graphs, implying that \(\mu_{c}<1\) for small enough sleeping rate. We also show that \(\mu_{c}<1\) for any sleeping rate in any vertex-transitive graph in which simple random walk has positive speed. Furthermore, we prove that \(\mu_{c}>0\) in any vertex-transitive amenable graph, and that \(\mu_{c}\in(0,1)\) for any sleeping rate on regular trees.

### MSC:

82C22 | Interacting particle systems in time-dependent statistical mechanics |

60K35 | Interacting random processes; statistical mechanics type models; percolation theory |

82C26 | Dynamic and nonequilibrium phase transitions (general) in statistical mechanics |

82C41 | Dynamics of random walks, random surfaces, lattice animals, etc. in time-dependent statistical mechanics |

82C27 | Dynamic critical phenomena in statistical mechanics |

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\textit{A. Stauffer} and \textit{L. Taggi}, Ann. Probab. 46, No. 4, 2190--2220 (2018; Zbl 1397.82038)

### References:

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