Math @ Duke

Publications [#329100] of Matthew S Junge
Papers Published
 Hoffman, C; Johnson, T; Junge, M, Recurrence and transience for the frog model on trees,
The Annals of Probability, vol. 45 no. 5
(September, 2017),
pp. 28262854, Institute of Mathematical Statistics [doi]
(last updated on 2019/08/07)
Abstract: © Institute of Mathematical Statistics, 2017. 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 dary tree. We prove the model undergoes a phase transition, finding it recurrent for d = 2 and transient for d ≥ 5. Simulations suggest strong recurrence for d = 2, weak recurrence for d = 3, and transience for d ≥ 4. Additionally, we prove a 01 law for all dary trees, and we exhibit a graph on which a 01 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 ≥ 6, which uses similar techniques but does not require computer assistance.


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