  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.
