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(d,p) on a d-ary tree with drift parameter p∈[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 pd 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 d≥2, pd≤1/3, achieving the best, universal upper bound predicted by the monotonicity conjecture.