## 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

### Keywords:

frog model; non-amenable; interacting random walk
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### References:

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