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 ${\sup }_{d\ge 3}{p}_d\le 1\mathrm{/}3$. We improve this bound to $5\mathrm{/}17$ with a shorter argument that generalizes to give bounds on ${\sup }_{d\ge m}{p}_d$. We additionally prove that $\mathrm{lim\hspace{0.17em}sup}{p}_d\le 1\mathrm{/}6$ by finding the limiting critical drift for a non-backtracking variant.