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April, 1980 Clustering and Dispersion Rates for Some Interacting Particle Systems on $\mathbb{Z}$
Maury Bramson, David Griffeath
Ann. Probab. 8(2): 183-213 (April, 1980). DOI: 10.1214/aop/1176994771


It is well known that the voter model on $\mathbb{Z}^d$ under initial product measure will converge weakly to a point process as $t \rightarrow \infty$; if $d \geqslant 3$, convergence will be to a nontrivial point process; if $d = 1,2$, convergence will be to a linear combination of trivial point processes. Therefore, for $d = 1,2$, the cluster size of a particular state around any fixed point tends to become arbitrarily large as $t \rightarrow \infty$. Here we examine the rate of growth for $d = 1$ of this clustering for the nearest neighbor voter model and the related problem of interparticle distance for nearest neighbor coalescing random walks and annihilating random walks. We show that under spatial renormalization $t^{\frac{1}{2}}$ these cluster sizes/interparticle distances in each case approach a nondegenerate distribution. We examine these distributions and obtain numerical estimates for these and related problems.


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Maury Bramson. David Griffeath. "Clustering and Dispersion Rates for Some Interacting Particle Systems on $\mathbb{Z}$." Ann. Probab. 8 (2) 183 - 213, April, 1980.


Published: April, 1980
First available in Project Euclid: 19 April 2007

zbMATH: 0429.60098
MathSciNet: MR566588
Digital Object Identifier: 10.1214/aop/1176994771

Primary: 60K35

Keywords: cluster size , Interacting particle system

Rights: Copyright © 1980 Institute of Mathematical Statistics

Vol.8 • No. 2 • April, 1980
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