Cover time for the frog model on trees
One frog is awake at the root of a d-ary tree of height n, and Poi(m) frogs are asleep at every other vertex. Awakened frogs move as simple random walks. When an awakened frog encounters a sleeping frog, the sleeping frog wakes up and starts its own random walk. I'll discuss two questions posed by Itai Benjamini: Is the cover time (the first time that all vertices have been visited) polynomial in n? And does the cover time undergo a phase transition in m? Our result is that the cover time is n log n if m is large, but is at least n^2 when m is small. Joint work with Christopher Hoffman and Matthew Junge.