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The frog model on non-amenable trees. (English) Zbl 1441.60084

Summary: We examine an interacting particle system on trees commonly referred to as the frog model. For its initial state, it begins with a single active particle at the root and i.i.d. \(\text{Poiss}(\lambda)\) many inactive particles at each non-root vertex. Active particles perform discrete time simple random walk and in the process activate any inactive particles they encounter. We show that for every non-amenable tree with bounded degree there exists a phase transition from transience to recurrence (with a non-trivial intermediate phase sometimes sandwiched in between) as \(\lambda\) varies.

MSC:

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

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