The frog model on trees with drift

TL;DR

This paper establishes an upper bound on the minimal drift needed for recurrence in the frog model on d-ary trees, introducing new couplings and simplifying recurrence proofs for binary trees.

Contribution

It introduces a coupling method across trees with different degrees and provides a simplified proof of recurrence for the frog model on binary trees, advancing understanding of critical drift.

Findings

Upper bound on minimal drift for recurrence on d-ary trees

Coupling techniques for frog models on nested graphs

Limit of critical drift as degree tends to infinity

Abstract

We provide a uniform upper bound on the minimal drift so that the one-per-site frog model on a $d$-ary tree is recurrent. To do this, we introduce a subprocess that couples across trees with different degrees. Finding couplings for frog models on nested sequences of graphs is known to be difficult. The upper bound comes from combining the coupling with a new, simpler proof that the frog model on a binary tree is recurrent when the drift is sufficiently strong. Additionally, we describe a coupling between frog models on trees for which the degree of the smaller tree divides that of the larger one. This implies that the critical drift has a limit as $d$ tends to infinity along certain subsequences.