On the minimal drift for recurrence in the frog model on $d$-ary trees

We study the recurrence of one-per-site frog model $\text{FM}(d, p)$ on a $d$-ary tree with drift parameter $p\in [0,1]$, which determines the bias of frogs' random walks. We are interested in the minimal drift $p_{d}$ so that the frog model is recurrent. Using a coupling argument together with a generating function technique, we prove that for all $d \ge 2$, $p_{d}\le 1/3$, which is the optimal universal upper bound.