Electronic Journal of Probability

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

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

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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

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

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)

frog model interacting random walks recurrence transience branching random walk percolation

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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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