Recurrence and transience for the frog model on trees

TL;DR

This paper investigates the recurrence and transience behavior of the frog model on infinite d-ary trees, establishing phase transitions at specific degrees and using a mix of analytical and computational methods.

Contribution

It proves the phase transition between recurrence and transience for the frog model on trees, including a recursive distributional equation and computer-assisted proofs.

Findings

Recurrent for d=2

Transient for d≥5

Phase transition at d=4 and 5

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 $d$-ary tree. We prove the model undergoes a phase transition, finding it recurrent for $d=2$ and transient for $d\geq 5$. Simulations suggest strong recurrence for $d=2$, weak recurrence for $d=3$, and transience for $d\geq 4$. 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 for $d \geq 6$, which uses similar techniques but does not require computer assistance.