Asymptotic behavior of the Brownian frog model

TL;DR

This paper studies an extension of the frog model in Euclidean space where particles perform Brownian motion and activate others upon contact, showing linear growth and Poisson distribution convergence under certain conditions.

Contribution

It introduces a new Euclidean-space Brownian frog model and proves its asymptotic linear expansion and Poisson convergence for subcritical radii.

Findings

Active particles form a linearly expanding ball

Active points approximate a Poisson process in fixed regions

Behavior depends on the radius being below the percolation threshold

Abstract

We introduce an extension of the frog model to Euclidean space and prove properties for the spread of active particles. Fix $r>0$ and place a particle at each point $x$ of a unit intensity Poisson point process $\mathcal P \subseteq \mathbb R^d - \mathbb B(0,r)$. Around each point in $\mathcal{P}$, put a ball of radius $r$. A particle at the origin performs Brownian motion. When it hits the ball around $x$ for some $x \in \mathcal P$, new particles begin independent Brownian motions from the centers of the balls in the cluster containing $x$. Subsequent visits to the cluster do nothing. This waking process continues indefinitely. For $r$ smaller than the critical threshold of continuum percolation, we show that the set of activated points in $\mathcal P$ approximates a linearly expanding ball. Moreover, in any fixed ball the set of active particles converges to a unit intensity Poisson point process.