Recurrence and Transience of Frogs with Drift on $\mathbb{Z}^d$

TL;DR

This paper investigates the recurrence and transience behavior of a frog model with drift on integer lattices, identifying parameter regimes that lead to either recurrence or transience in different dimensions.

Contribution

It introduces a two-parameter frog model with drift, providing conditions for recurrence and transience, and highlights differences between two and higher dimensions.

Findings

Recurrence occurs under certain parameter conditions in $d=2$.

Transience is established for other parameter regimes in $d geq 2$.

Different behaviors are observed between $d=2$ and $d geq 3$.

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.