%0 Journal Article
%T Recurrence and transience for the frog model on trees
%A Christopher Hoffman
%A Tobias Johnson
%A Matthew Junge
%J Mathematics
%D 2014
%I arXiv
%X 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.
%U http://arxiv.org/abs/1404.6238v5