Recurrence and transience for the frog model on trees

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 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.