RECURRENCE AND TRANSIENCE FOR THE FROG MODEL ON TREES

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 >= 5. Simulations suggest strong recurrence for d = 2, weak recurrence for d = 3, and transience for d >= 4. 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 ford >= 6, which uses similar techniques but does not require computer assistance.