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

Abstract

We study the recurrence property of one-per-site frog model $FM(\mathit{d},\mathit{p})$ on a d-ary tree with drift parameter $\mathit{p}\in [0,1]$, which determines the bias of frogs’ random walks. In this model, active frogs move toward the root with probability p or otherwise move to a uniformly chosen child vertex. Whenever a site is visited for the first time, a new active frog is introduced at the site. We are interested in the minimal drift ${\mathit{p}}_{\mathit{d}}$ so that the frog model is recurrent. Using a coupling argument together with a recursive construction of two series of polynomials involved in the generating functions, we prove that for all $\mathit{d}\ge 2$, ${\mathit{p}}_{\mathit{d}}\le 1/3$, achieving the best, universal upper bound predicted by the monotonicity conjecture.