Critical drift estimates for the frog model on trees

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≥3p_d ≤1/3. We improve this bound to 5∕17 with a shorter argument that generalizes to give bounds on sup_d≥3p_d . We additionally prove that limsuppp_d ≤ 1∕6 by finding the limiting critical drift for a non-backtracking variant.