The Annals of Probability

Recurrence and transience for the frog model on trees

Christopher Hoffman, Tobias Johnson, and Matthew Junge

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Abstract

The frog model is a growing system of random walks where a particle is added whenever a new site is visited. A longstanding open question is how often the root is visited on the infinite $d$-ary tree. We prove the model undergoes a phase transition, finding it recurrent for $d=2$ and transient for $d\geq5$. Simulations suggest strong recurrence for $d=2$, weak recurrence for $d=3$, and transience for $d\geq4$. Additionally, we prove a 0–1 law for all $d$-ary trees, and we exhibit a graph on which a 0–1 law does not hold.

To prove recurrence when $d=2$, we construct a recursive distributional equation for the number of visits to the root in a smaller process and show the unique solution must be infinity a.s. The proof of transience when $d=5$ relies on computer calculations for the transition probabilities of a large Markov chain. We also include the proof for $d\geq 6$, which uses similar techniques but does not require computer assistance.

Article information

Source
Ann. Probab., Volume 45, Number 5 (2017), 2826-2854.

Dates
Received: June 2015
Revised: May 2016
First available in Project Euclid: 23 September 2017

Permanent link to this document
https://projecteuclid.org/euclid.aop/1506132027

Digital Object Identifier
doi:10.1214/16-AOP1125

Mathematical Reviews number (MathSciNet)
MR3706732

Zentralblatt MATH identifier
1385.60058

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 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 zero–one law

Citation

Hoffman, Christopher; Johnson, Tobias; Junge, Matthew. Recurrence and transience for the frog model on trees. Ann. Probab. 45 (2017), no. 5, 2826--2854. doi:10.1214/16-AOP1125. https://projecteuclid.org/euclid.aop/1506132027


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Supplemental materials

  • Computer code and corresponding explanation for the proofs of Proposition 19 and Theorem 1(ii). We provide the source code referred to in the proofs of Proposition 19 and Theorem 1(ii), as well as some documentation.