Electronic Journal of Probability

Asymptotic behavior of the Brownian frog model

Erin Beckman, Emily Dinan, Rick Durrett, Ran Huo, and Matthew Junge

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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.

Article information

Electron. J. Probab., Volume 23 (2018), paper no. 104, 19 pp.

Received: 15 December 2017
Accepted: 21 August 2018
First available in Project Euclid: 20 October 2018

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Zentralblatt MATH identifier

Primary: 60J25: Continuous-time Markov processes on general state spaces

frog model shape theorem continuum percolation

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Beckman, Erin; Dinan, Emily; Durrett, Rick; Huo, Ran; Junge, Matthew. Asymptotic behavior of the Brownian frog model. Electron. J. Probab. 23 (2018), paper no. 104, 19 pp. doi:10.1214/18-EJP215. https://projecteuclid.org/euclid.ejp/1540000928

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