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 ${\sup }_{d\ge m}{p}_d$ and prove monotonicity of critical values associated to a self-similar variant.