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 is an element of [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 p(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 d >= 2, p(d) <= 1/3, achieving the best, universal upper bound predicted by the monotonicity conjecture.