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

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

doi:10.1214/18-EJP216

2018 Recurrence and transience of frogs with drift on $\mathbb{Z} ^d$

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

Electron. J. Probab. 23: 1-23 (2018). 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.

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Christian Döbler, Nina Gantert, Thomas Höfelsauer, Serguei Popov, and Felizitas Weidner "Recurrence and transience of frogs with drift on $\mathbb{Z} ^d$," Electronic Journal of Probability 23(none), 1-23, (2018). https://doi.org/10.1214/18-EJP216

Received: 1 September 2017; Accepted: 22 August 2018; Published: 2018

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