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February 2001 Asymptotic density in a threshold coalescing and annihilating random walk
David Stephenson
Ann. Probab. 29(1): 137-175 (February 2001). DOI: 10.1214/aop/1008956326


We consider an interacting random walk on $\mathbb{Z}^d$ where particles interact only at times when a particle jumps to a site at which there are at least $n - 1$ other particles present. If there are $i \ge n - 1$ particles present, then the particle coalesces (is removed from the system) with probability $c_i$ and annihilates (is removed along with another particle) with probability $a_i$. We call this process the $n$-threshold randomly coalescing and annihilating random walk. We show that, for $n \ge 3$, if both $a_i$ and $a_i + c_i$ are increasing in $i$ and if the dimension $d$ is at least $2n + 4$, then $$P\{\text{the origin is occupied at time $t$}\}\sim C(d, n) t^{−\frac{1}{n-1}},\\ E\{\text{number of particles at the origin at time $t$}\} \sim C(d, n) t^{−\frac{1}{n-1}}.$$ The constants $C(d, n)$ are explicitly identified. The proof is an extension of a result obtained by Kesten and van den Berg for the 2-threshold coalescing random walk and is based on an approximation for $dE(t)/dt$.


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David Stephenson. "Asymptotic density in a threshold coalescing and annihilating random walk." Ann. Probab. 29 (1) 137 - 175, February 2001.


Published: February 2001
First available in Project Euclid: 21 December 2001

zbMATH: 1031.60086
MathSciNet: MR1825146
Digital Object Identifier: 10.1214/aop/1008956326

Primary: 60K35 , 82C22

Keywords: annihilating random walk , Coealescing random walk , particle density , threshold

Rights: Copyright © 2001 Institute of Mathematical Statistics


Vol.29 • No. 1 • February 2001
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