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November, 1983 Hydrodynamics of the Voter Model
Errico Presutti, Herbert Spohn
Ann. Probab. 11(4): 867-875 (November, 1983). DOI: 10.1214/aop/1176993437


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 $m(q, t)$.


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Errico Presutti. Herbert Spohn. "Hydrodynamics of the Voter Model." Ann. Probab. 11 (4) 867 - 875, November, 1983.


Published: November, 1983
First available in Project Euclid: 19 April 2007

zbMATH: 0527.60094
MathSciNet: MR714951
Digital Object Identifier: 10.1214/aop/1176993437

Primary: 60K35
Secondary: 82A05

Keywords: fluctuations of the magnetization field , states of local equilibrium , voter model

Rights: Copyright © 1983 Institute of Mathematical Statistics

Vol.11 • No. 4 • November, 1983
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