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

24 pages, 8 figures to appear in Annals of Probability

References in corpus (5)

• A survey of random processes with reinforcement
• From transience to recurrence with Poisson tree frogs
• Activated Random Walkers: Facts, Conjectures and Challenges
• Absorbing-State Phase Transition for Driven-Dissipative Stochastic Dynamics on $Z$
• Recurrence for the frog model with drift on ${\mathbb{Z}}^d$

Cited by in corpus (16)

• From transience to recurrence with Poisson tree frogs
• A new upper bound for the critical probability of the frog model on homogeneous trees
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• Cover time for the frog model on trees
• Sensitivity of the frog model to initial conditions
• Rapid social connectivity
• Laws of large numbers for the frog model on the complete graph
• Deviation bounds for the first passage time in the frog model
• The continuous-time frog model can spread arbitrarily fast
• Active Phase for Activated Random Walks on the Lattice in all Dimensions
• Critical parameter of the frog model on homogeneous trees with geometric lifetime
• A stochastic combustion model with thresholds on trees
• Critical percolation and A+B --> 2A dynamics
• The asymptotic shape theorem for the frog model on finitely generated abelian groups