Critical drift estimates for the frog model on trees

Abstract

Place an active particle at the root of a d-ary tree and a single dormant particle at each non-root site. In discrete time, active particles move towards the root with probability p and, otherwise, away from the root to a uniformly sampled child vertex. When an active particle moves to a site containing a dormant particle, the dormant particle becomes active. The critical drift $p_d$ is the infimum over all p for which infinitely many particles visit the root almost surely. Guo, Tang, and Wei proved that ${sup}_{d\ge 3}{p}_d\le 1∕3$. We improve this bound to $5∕17$ with a shorter argument that generalizes to give bounds on ${sup}_{d\ge m}{p}_d$. We additionally prove that $lim\hspace{0.17em}sup{p}_d\le 1∕6$ by finding the limiting critical drift for a non-backtracking variant.