## 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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