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

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 bias frogs’ random walks. In this model, active frogs move toward root probability p or otherwise to uniformly chosen child vertex. Whenever site is visited for first time, new introduced at site. are interested in minimal pd so that recurrent. Using coupling argument together recursive construction two series polynomials involved generating functions, we prove all d≥2, pd≤1/3, achieving best, universal upper bound predicted by monotonicity conjecture.