Critical drift estimates for the frog model on trees

Place an active particle at the root of ad-ary tree and a single dormant particle a teach 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 >= 3) p(d )<= 1/3. We improve this bound to 5/17with a shorter argument that generalizes to give bounds on sup(d >= m)p(d). We additionally prove that lim sup p(d )<= 1/6 by finding the limiting critical drift for a non-backtracking variant.