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December, 1981 Limiting Point Processes for Rescalings of Coalescing and Annihilating Random Walks on $Z^d$
Richard Arratia
Ann. Probab. 9(6): 909-936 (December, 1981). DOI: 10.1214/aop/1176994264

Abstract

Let $p(x, y)$ be an arbitrary random walk on $Z^d$. Let $\xi_t$ be the system of coalescing random walks based on $p$, starting with all sites occupied, and let $\eta_t$ be the corresponding system of annihilating random walks. The spatial rescalings $P(0 \in \xi_t)^{1/d}\xi_t$ for $t \geqq 0$ form a tight family of point processes on $R^d$. Any limiting point process as $t \rightarrow\infty$ has Lesbegue measure as its intensity, and has no multiple points. When $p$ is simple random walk on $Z^d$ these rescalings converge in distribution, to the simple Poisson point process for $d \geq 2$, and to a non-Poisson limit for $d = 1$. For a large class of $p$, we prove that $P(0 \in \eta_t)/P(0 \in \xi_t) \rightarrow 1/2$ as $t \rightarrow\infty$. A generalization of this result, proved for nearest neighbor random walks on $Z^1$, and for all multidimensional $p$, implies that the limiting point process for rescalings $P(0 \in \xi_t)^{1/d}\eta_t$ of the system of annihilating random walks is the one half thinning of the limiting point process for the corresponding coalescing system.

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Richard Arratia. "Limiting Point Processes for Rescalings of Coalescing and Annihilating Random Walks on $Z^d$." Ann. Probab. 9 (6) 909 - 936, December, 1981. https://doi.org/10.1214/aop/1176994264

Information

Published: December, 1981
First available in Project Euclid: 19 April 2007

zbMATH: 0496.60098
MathSciNet: MR632966
Digital Object Identifier: 10.1214/aop/1176994264

Subjects:
Primary: 60K35

Rights: Copyright © 1981 Institute of Mathematical Statistics

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Vol.9 • No. 6 • December, 1981
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