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.
Christopher Hoffman. Tobias Johnson. Matthew Junge. "From transience to recurrence with Poisson tree frogs." Ann. Appl. Probab. 26 (3) 1620 - 1635, June 2016. https://doi.org/10.1214/15-AAP1127