Asymptotic behavior of the brownian frog model

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 P⊆ℝd−B(0,r). Around each point in 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 ∈ 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 P approximates a linearly expanding ball. Moreover, in any fixed ball the set of active particles converges to a unit intensity Poisson point process.

Duke Scholars

Author Richard Timothy Durrett Mathematics