## Electronic Journal of Probability

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

#### 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.

#### Article information

Source
Electron. J. Probab., Volume 23 (2018), paper no. 88, 23 pp.

Dates
Received: 1 September 2017
Accepted: 22 August 2018
First available in Project Euclid: 12 September 2018

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1536717747

Digital Object Identifier
doi:10.1214/18-EJP216

Mathematical Reviews number (MathSciNet)
MR3858916

Zentralblatt MATH identifier
06964782

#### Citation

Döbler, Christian; Gantert, Nina; Höfelsauer, Thomas; Popov, Serguei; Weidner, Felizitas. Recurrence and transience of frogs with drift on $\mathbb{Z} ^d$. Electron. J. Probab. 23 (2018), paper no. 88, 23 pp. doi:10.1214/18-EJP216. https://projecteuclid.org/euclid.ejp/1536717747

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