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

Abstract

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.

21 pages, 2 figures