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

TL;DR

This paper investigates the minimal bias in frog models on d-ary trees that guarantees recurrence, establishing a universal upper bound of 1/3 for all degrees d.

Contribution

It introduces a coupling and generating function approach to prove the universal upper bound on minimal drift for recurrence in frog models on d-ary trees.

Findings

Universal upper bound p_d ≤ 1/3 for all d ≥ 2

Optimality of the 1/3 bound across different degrees

Methodology applicable to similar stochastic processes

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.