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Asymptotic behavior of the Brownian frog model. (English) Zbl 1402.60124

Summary: 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.

MSC:

60K35 Interacting random processes; statistical mechanics type models; percolation theory
60J25 Continuous-time Markov processes on general state spaces
60J55 Local time and additive functionals
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