**The frog model on trees.**

Fix a graph G and place some number (random or otherwise) of sleeping frogs at each site, as well as one awake frog at the root. Set things in motion by having awake frogs perform independent simple random walk, waking any "sleepers" they encounter. Say the model is recurrent if the root is a.s. visited by infinitely many frogs and otherwise transient. When G is the rooted d-ary tree with one-frog-per-site we prove a phase transition from recurrence to transience as d increases. Alternatively, for fixed d with Poi(m)-frogs-per-site we prove a phase transition from transience to recurrence as m increases. The proofs use two different recursions and two different versions of stochastic domination. Several open problems will be discussed. Joint with Christopher Hoffman and Tobias Johnson.

Speaker: Matthew Junge is a 5th year PhD student at the University of Washington with advisor Christopher Hoffman. He received his masters in algebraic geometry also at the University of Washington under James Morrow and William McGovern.