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# Asymptotic behavior of the Brownian frog model

 dc.contributor.author Beckman, Erin dc.contributor.author Huo, Ran dc.contributor.author Dinan, Emily dc.contributor.author Durrett, Rick dc.contributor.author Junge, Matthew dc.date.accessioned 2018-10-21T02:46:03Z dc.date.available 2018-10-21T02:46:03Z dc.identifier.uri https://hdl.handle.net/10161/17596 dc.description.abstract We introduce an extension of the frog model to Euclidean space and prove properties for the spread of active particles. The new geometry introduces a phase transition that does not occur for the frog model on the lattice. 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 sites in $\mathcal P$ behaves like a linearly expanding ball. Moreover, in any fixed ball the set of active particles converges to a unit intensity Poisson point process. Lastly, we prove that the model expands at rate at least $t^{2- \epsilon}$ when $d=2$ and $r$ equals the critical threshold in continuum percolation. dc.subject math.PR dc.subject math.PR dc.subject 60J25 dc.title Asymptotic behavior of the Brownian frog model dc.type Journal article dc.date.updated 2018-10-21T02:46:02Z pubs.organisational-group Student pubs.organisational-group Duke pubs.organisational-group Mathematics pubs.organisational-group Trinity College of Arts & Sciences
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