Open Access
April, 1986 Diffusive Clustering in the Two Dimensional Voter Model
J. Theodore Cox, David Griffeath
Ann. Probab. 14(2): 347-370 (April, 1986). DOI: 10.1214/aop/1176992521


We study the behavior of an interacting particle system known as the voter model in two dimensions. This process provides a simple example of "critical clustering" among two colors, say green and black, in the plane. The paper begins with some computer simulations, and a survey of known results concerning the voter model in the three qualitatively distinct cases: three or more dimensions (high), one dimension (low), and two dimensions (critical). Our main theorem, for the planar model, states roughly that at large times $t$ the proportion of green sites on a box of side $t^{\alpha/2}$ centered at the origin fluctuates with $\alpha$ according to a time change of the Fisher-Wright diffusion. Some applications of the theorem, and several related results, are described.


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J. Theodore Cox. David Griffeath. "Diffusive Clustering in the Two Dimensional Voter Model." Ann. Probab. 14 (2) 347 - 370, April, 1986.


Published: April, 1986
First available in Project Euclid: 19 April 2007

zbMATH: 0658.60131
MathSciNet: MR832014
Digital Object Identifier: 10.1214/aop/1176992521

Primary: 60K35

Keywords: clustering , Coalescing random walks , exchangeable random field , genetics diffusion , Infinite particle system , voter model

Rights: Copyright © 1986 Institute of Mathematical Statistics

Vol.14 • No. 2 • April, 1986
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