The frog model on trees with drift

Erin Beckman,Matthew Junge,Si Tang,Natalie Frank,Yufeng Jiang

doi:10.1214/19-ecp235

Abstract

We provide a uniform upper bound on the minimal drift so that the one-per-site frog model on a $d$-ary tree is recurrent. To do this, we introduce a subprocess that couples across trees with different degrees. Finding couplings for frog models on nested sequences of graphs is known to be difficult. The upper bound comes from combining the coupling with a new, simpler proof that the frog model on a binary tree is recurrent when the drift is sufficiently strong. Additionally, we describe a coupling between frog models on trees for which the degree of the smaller tree divides that of the larger one. This implies that the critical drift has a limit as $d$ tends to infinity along certain subsequences.

Highlights

We provide a uniform upper bound on the minimal drift so that the one-per-site frog model on a d-ary tree is recurrent
We describe a coupling between frog models on trees for which the degree of the smaller tree divides that of the larger one
We study the one-per-site frog model with drift on the rooted d-ary tree Td

Summary

Introduction

We study the one-per-site frog model with drift on the rooted d-ary tree Td. Initially there is a single awake frog at the root and one sleeping frog at each non-root vertex. It was first studied by Gantert and Schmidt with i.i.d η frogs per site and a drift in the e1 direction on Z [GS09] They showed that the process is recurrent if and only if E log+ η = ∞ regardless of the drift. Döbler and Pfeifroth showed that the frog model is recurrent on Zd for d ≥ 2 so long as E log(+d+1)/2 η = ∞ [DP14] It was open for some time whether, unlike the d = 1 case, there is a phase transition as the drift is varied. Follow-up work by Hoffman, Johnson, and Junge showed that the frog model with unbiased random walks is recurrent for any d so long as Ω(d) sleeping frogs are placed at each site [HJJ16, JJ16a, JJ16b].

This result together with the lower bound imply

Since p