The Annals of Probability

Diffusive Clustering in the Two Dimensional Voter Model

J. Theodore Cox and David Griffeath

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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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Ann. Probab., Volume 14, Number 2 (1986), 347-370.

First available in Project Euclid: 19 April 2007

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Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Infinite particle system voter model coalescing random walks genetics diffusion clustering exchangeable random field


Cox, J. Theodore; Griffeath, David. Diffusive Clustering in the Two Dimensional Voter Model. Ann. Probab. 14 (1986), no. 2, 347--370. doi:10.1214/aop/1176992521.

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