Cover time for the frog model on trees

TL;DR

This paper analyzes the cover time of the frog model on trees, showing how initial particle density influences whether the entire tree is visited quickly or takes exponentially long.

Contribution

It provides a precise characterization of the cover time behavior for the frog model on trees based on initial particle density.

Findings

High initial density ($ o ext{Omega}(d^2)$) leads to cover time of $ heta(n ext{log} n)$

Low initial density ($ o O(d)$) results in exponential cover time $ ext{exp}( heta( extsqrt{n}))$

The phase transition depends on the initial particle density relative to the degree of the tree.

Abstract

The frog model is a branching random walk on a graph in which particles branch only at unvisited sites. Consider an initial particle density of $\mu$ on the full $d$-ary tree of height $n$. If $\mu= \Omega( d^2)$, all of the vertices are visited in time $\Theta(n\log n)$ with high probability. Conversely, if $\mu = O(d)$ the cover time is $\exp(\Theta(\sqrt n))$ with high probability.