Recurrence and transience for the frog model on trees

By Christopher E. Hoffman, Tobias Johnson, Matthew Junge. 2014

68 citations recorded by the source corpus.

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 dd-ary tree. We prove the model undergoes a phase transition, finding it recurrent for d=2d=2 and transient for d≥5d≥5. Simulations suggest strong recurrence for d=2d=2, weak recurrence for d=3d=3, and transience for d≥4d≥4. Additionally, we prove a 0–1 law for all dd-ary trees, and we exhibit a graph on which a 0–1 law does not hold. To prove recurrence when d=2d=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=5d=5 relies on computer calculations for the transition probabilities of a large Markov chain. We also include the proof for d≥6d≥6, which uses similar techniques but does not require computer assistance.

Subfields: Condensed Matter Physics, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/e3e79b36087d3ef7d090bf4f4c91899afa547eb3

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In Condensed Matter Physics

In Mathematical Physics

In Statistics and Probability

Citations in the corpus

From transience to recurrence with Poisson tree frogs. (2015, methodology): “Our proof in [HJJ14] instead uses an exotic definition of stochastic dominance, where π1 is dominated by π2 if the probability generating function of π1 is greater than the probability generating function of π2.”