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January 2000 Asymptotic density in a coalescing random walk model
Harry Kesten, J. van den Berg
Ann. Probab. 28(1): 303-352 (January 2000). DOI: 10.1214/aop/1019160121


We consider a system of particles,each of which performs a continuous time random walk on $\mathbb{Z}^d$ . The particles interact only at times when a particle jumps to a site at which there are a number of other particles present. If there are $j$ particles present, then the particle which just jumped is removed from the system with probability $p_j$. We show that if $p_j$ is increasing in $j$ and if the dimension $d$ is at least 6 and if we start with one particle at each site of $\mathbb{Z}^d$, then $p(t):= P\{there is at least one particle at the origin at time t\}\sim C(d)/t$. The constant $C(d)$ is explicitly identified. We think the result holds for every dimension $d \geq 3$ and we briefly discuss which steps in our proof need to be sharpened to weaken our assumption $d \geq 6$.

The proof is based on a justification of a certain mean field approximation for $dp(t)/dt$.The method seems applicable to many more models of coalescing and annihilating particles.


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Harry Kesten. J. van den Berg. "Asymptotic density in a coalescing random walk model." Ann. Probab. 28 (1) 303 - 352, January 2000.


Published: January 2000
First available in Project Euclid: 18 April 2002

zbMATH: 1044.60098
MathSciNet: MR1756007
Digital Object Identifier: 10.1214/aop/1019160121

Primary: 60K35
Secondary: 70J15

Keywords: asymptotic particle density , coalescing random walk , method of bounded differences

Rights: Copyright © 2000 Institute of Mathematical Statistics


Vol.28 • No. 1 • January 2000
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