## The Annals of Probability

### Diffusive Clustering in the Two Dimensional Voter Model

#### Abstract

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.

#### Article information

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

Dates
First available in Project Euclid: 19 April 2007

https://projecteuclid.org/euclid.aop/1176992521

Digital Object Identifier
doi:10.1214/aop/1176992521

Mathematical Reviews number (MathSciNet)
MR832014

Zentralblatt MATH identifier
0658.60131

JSTOR