The Poisson frog model on Galton-Watson trees

We consider an interacting particle system on trees known as the frog model: initially, a single active particle begins at the root and i.i.d. Poisson(u) many inactive particles are placed at each non-root vertex.  Active particles perform discrete time simple random walk and activate the inactive particles they encounter.  It has been shown by Hoffman, Johnson, and Junge that on regular trees, there is a critical value u_c separating recurrent and transient regimes.  Little is known, however, about the behavior of the frog model on random structures, and other graphs that do not posses a high degree of self-similarity.  In this talk, I'll discuss our recent results showing that for Galton-Watson trees with certain types of offspring distributions there does exist a critical value u_c separating recurrent and transient regimes for almost surely every tree, thereby partially answering a question of Hoffman-Johnson-Junge.  I'll also discuss a related proof showing that for every non-amenable tree with bounded degree there exists a phase transition from transience to recurrence (with a non-trivial intermediate phase sometimes sandwiched in between) as u varies.  This is based on joint work with Marcus Michelen.