Improved critical drift estimates for the frog model on trees

Place an active particle at the root of the infinite d-ary tree and dormant particles at each non-root site. Active particles move towards the root with probability p and otherwise move to a uniformly sampled child vertex. When an active particle moves to a site containing dormant particles, all the particles at the site become active. The critical drift p_d is the infimum over all p for which infinitely many particles visit the root almost surely. We give improved bounds on sup_d≥mp_d and prove monotonicity of critical values associated to a self-similar variant.