From transience to recurrence with Poisson tree frogs

Abstract

Consider the following interacting particle system on the $d$-ary tree, known as the frog model: Initially, one particle is awake at the root and i.i.d. Poisson many particles are sleeping at every other vertex. Particles that are awake perform simple random walks, awakening any sleeping particles they encounter. We prove that there is a phase transition between transience and recurrence as the initial density of particles increases, and we give the order of the transition up to a logarithmic factor.

arXiv commentPublished at http://dx.doi.org/10.1214/15-AAP1127 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)

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• On the range of the transient frog model on Z
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• Laws of large numbers for the frog model on the complete graph
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• The continuous-time frog model can spread arbitrarily fast
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• A stochastic combustion model with thresholds on trees
• A zero-one law for recurrence and transience of frog processes
• The frog model on Galton-Watson trees
• On the minimal drift for recurrence in the frog model on $d$-ary trees
• Passage time of the frog model has a sublinear variance