## The Annals of Probability

### 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 $m(q, t)$.

#### Article information

Source
Ann. Probab., Volume 11, Number 4 (1983), 867-875.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176993437

Digital Object Identifier
doi:10.1214/aop/1176993437

Mathematical Reviews number (MathSciNet)
MR714951

Zentralblatt MATH identifier
0527.60094

JSTOR