Infection spread for the frog model on trees

TL;DR

This paper analyzes the infection spread in the frog model on rooted d-ary trees, demonstrating linear growth in infected sites and visits to the root when particle density is sufficiently high.

Contribution

It establishes that with high particle density, the infected region expands linearly and root visits grow proportionally, advancing understanding of infection dynamics on trees.

Findings

Visited sites form a linearly expanding ball.

Number of visits to the root grows linearly.

Results hold for particle density ^2 or higher.

Abstract

The frog model is an infection process in which dormant particles begin moving and infecting others once they become infected. We show that on the rooted $d$-ary tree with particle density $\Omega(d^2)$, the set of visited sites contains a linearly expanding ball and the number of visits to the root grows linearly with high probability.