From transience to recurrence with Poisson tree frogs

TL;DR

This paper studies a frog model on a d-ary tree, showing a phase transition from transience to recurrence as the initial particle density increases, with detailed analysis of the transition's order.

Contribution

It establishes the existence and nature of a phase transition in the frog model on trees, providing bounds on the transition's order.

Findings

Identifies a phase transition between transience and recurrence.

Provides bounds on the order of the phase transition.

Analyzes the impact of initial particle density on system behavior.

Abstract

Consider the following interacting particle system on the $d$-ary tree, known as the frog model: Initially, one particle is awake at the root and i.i.d. Poisson many particles are sleeping at every other vertex. Particles that are awake perform simple random walks, awakening any sleeping particles they encounter. We prove that there is a phase transition between transience and recurrence as the initial density of particles increases, and we give the order of the transition up to a logarithmic factor.