The frog model on non-amenable trees

TL;DR

This paper studies the frog model on non-amenable trees, revealing a phase transition from transience to recurrence as the initial particle density varies, with implications for understanding particle activation dynamics.

Contribution

It establishes the existence of a phase transition on all non-amenable trees with bounded degree, including intermediate phases, which was previously unknown.

Findings

Phase transition from transience to recurrence as λ varies

Existence of non-trivial intermediate phases

Applicable to all non-amenable trees with bounded degree

Abstract

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. $\mathrm{Poiss}(\lambda)$ 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 $\textit{every}$ 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.