Cover time for the frog model on trees

Abstract

The frog model is a branching random walk on a graph in which particles branch only at unvisited sites. Consider an initial particle density of $\mu$ on the full $d$-ary tree of height $n$. If $\mu =\Omega (d^2)$, all of the vertices are visited in time $\Theta (n\log n)$ with high probability. Conversely, if $\mu =O(d)$ the cover time is $\exp (\Theta (\sqrt{n}))$ with high probability.

36 pages; revisions in response to referees' comments; accepted in Forum of Math Sigma, Probability