The critical density for the frog model is the degree of the tree

TL;DR

This paper investigates the frog model on rooted d-ary trees, establishing that the phase transition from transience to recurrence occurs at a critical density proportional to the tree's degree.

Contribution

The authors prove that the critical density for the frog model's phase transition scales with the degree of the tree, providing a precise understanding of the transition point.

Findings

Critical density is proportional to the degree of the tree.

Transition from transience to recurrence occurs at this critical density.

The result clarifies the relationship between tree degree and frog model behavior.

Abstract

The frog model on the rooted d-ary tree changes from transient to recurrent as the number of frogs per site is increased. We prove that the location of this transition is on the same order as the degree of the tree.