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.