The Annals of Applied Probability

From transience to recurrence with Poisson tree frogs

Christopher Hoffman, Tobias Johnson, and Matthew Junge

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Abstract

Consider the following interacting particle system on the $d$-ary tree, known as the frog model: Initially, one particle is awake at the root and i.i.d. Poisson many particles are sleeping at every other vertex. Particles that are awake perform simple random walks, awakening any sleeping particles they encounter. We prove that there is a phase transition between transience and recurrence as the initial density of particles increases, and we give the order of the transition up to a logarithmic factor.

Article information

Source
Ann. Appl. Probab., Volume 26, Number 3 (2016), 1620-1635.

Dates
Received: January 2015
Revised: June 2015
First available in Project Euclid: 14 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1465905013

Digital Object Identifier
doi:10.1214/15-AAP1127

Mathematical Reviews number (MathSciNet)
MR3513600

Zentralblatt MATH identifier
1345.60116

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

Keywords
Frog model transience recurrence phase transition

Citation

Hoffman, Christopher; Johnson, Tobias; Junge, Matthew. From transience to recurrence with Poisson tree frogs. Ann. Appl. Probab. 26 (2016), no. 3, 1620--1635. doi:10.1214/15-AAP1127. https://projecteuclid.org/euclid.aoap/1465905013


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