Hydrodynamics of the Voter Model

Abstract

We study the voter model on $\mathbb{Z}^d, d \geqq 3$, for a sequence $\mu^\varepsilon$ of initial states which have a gradient in the mean magnetization of the order $\varepsilon, \varepsilon \rightarrow 0$. We prove that the magnetization field $m^\varepsilon(f, t) = \varepsilon^d \sum f(\varepsilon x)\eta(x, \varepsilon^{-2}t)$ tends to a deterministic field $m(f, t) = \int dqf(q)m(q, t)$ as $\varepsilon \rightarrow 0. m(q, t)$ is the solution of the diffusion equation. The fluctuations of $m^\varepsilon(f, t)$ around its mean are given by an infinite dimensional, non-homogeneous Ornstein-Uhlenbeck process. In the limit as $\varepsilon \rightarrow 0$, locally, i.e. around $(\varepsilon^{-1}q, \varepsilon^{-2}t)$, the voter model is in equilibrium with parameter $m(q, t)$.