Recurrence and transience of frogs with drift on $\mathbb{Z} ^d$

Thomas Höfelsauer,Serguei Popov,Nina Gantert,Christian Döbler,Felizitas Weidner

doi:10.1214/18-ejp216

Abstract

We study the frog model on $\mathbb{Z}^d$ with drift in dimension $d \geq 2$ and establish the existence of transient and recurrent regimes depending on the transition probabilities. We focus on a model in which the particles perform nearest neighbour random walks which are balanced in all but one direction. This gives a model with two parameters. We present conditions on the parameters for recurrence and transience, revealing interesting differences between dimension $d=2$ and dimension $d \geq 3$. Our proofs make use of (refined) couplings with branching random walks for the transience, and with percolation for the recurrence.

Highlights

Introduction and main resultsThe frog model is a model of interacting random walks or, more generally, Markov chains on a graph G = (V, E) in discrete time N0
The frog model with V = Zd, E the set of nearest-neighbour edges on Zd, x0 := 0, ηx = 1 for each x ∈ Zd \ {0} and the underlying random walk being simple random walk (SRW) on Zd was studied by Telcs and Wormald [19] who, called it egg model
They proved that the frog model on Zd with underlying random walk which has an arbitrary drift to the right is recurrent provided that

Summary

Introduction and main results

The recurrence part of the latter result was generalised to any dimension d by Döbler and Pfeifroth in [4] They proved that the frog model on Zd with underlying (irreducible) random walk which has an arbitrary drift to the right is recurrent provided that. In dimension d = 2 the frog model is recurrent whenever α or w are sufficiently small, i.e. if the underlying transition mechanism is almost balanced It is transient for α or w close to 1. For a discussion about their shape we refer the reader to Conjecture 4.1 at the end of this paper These results show that, in contrast to d = 1, recurrence and transience do depend on the drift in every dimension d ≥ 2.

Preliminaries

Proofs

Open problems

79. MR-3283610