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.
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