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 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.
Highlights
The frog model is a particular system of interacting random walks on a rooted graph
It starts with a single active particle at the root, and some collection of inactive particles distributed among the non-root vertices
In a recent work [10], the authors studied the frog model with Poiss(λ) frogs per vertex on Galton-Watson trees; under mild assumptions on the offspring distribution, we proved that there is a sharp transition Galton-Watson almost-surely from transience to recurrence as λ varies
Summary
The frog model is a particular system of interacting random walks on a rooted graph. It starts with a single active particle at the root, and some collection of inactive particles distributed among the non-root vertices. Active particles perform mutually independent discrete-time simple random walk, and any time an active particle meets a group of inactive particles, the inactive particles become active In this system the particles are often referred to as “frogs,” where active particles are considered “awake” and inactive particles “sleeping.” For infinite graphs, studies of the frog model often involve establishing whether it is recurrent (meaning almost surely infinitely many active particles hit the root) or transient (meaning almost surely only finitely many active particles ever hit the root). In a recent work [10], the authors studied the frog model with Poiss(λ) frogs per vertex on Galton-Watson trees; under mild assumptions on the offspring distribution, we proved that there is a sharp transition Galton-Watson almost-surely from transience to recurrence as λ varies. The goal of this work is to continue to break free from the restriction of quasi-transitivity and prove the existence of a phase transition for a wide class of deterministic trees
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
