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September 2017 Recurrence and transience for the frog model on trees
Christopher Hoffman, Tobias Johnson, Matthew Junge
Ann. Probab. 45(5): 2826-2854 (September 2017). DOI: 10.1214/16-AOP1125

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.

Citation

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Christopher Hoffman. Tobias Johnson. Matthew Junge. "Recurrence and transience for the frog model on trees." Ann. Probab. 45 (5) 2826 - 2854, September 2017. https://doi.org/10.1214/16-AOP1125

Information

Received: 1 June 2015; Revised: 1 May 2016; Published: September 2017
First available in Project Euclid: 23 September 2017

zbMATH: 1385.60058
MathSciNet: MR3706732
Digital Object Identifier: 10.1214/16-AOP1125

Subjects:
Primary: 60K35
Secondary: 60J10, 60J80

Rights: Copyright © 2017 Institute of Mathematical Statistics

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Vol.45 • No. 5 • September 2017
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