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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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References:

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