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November, 1983 Occupation Time Limit Theorems for the Voter Model
J. Theodore Cox, David Griffeath
Ann. Probab. 11(4): 876-893 (November, 1983). DOI: 10.1214/aop/1176993438


Let $\{\eta^\theta_s(x)\}, s \geq 0, x \in Z^d$ be the basic voter model starting from product measure with density $\theta(0 < \theta < 1).$ We consider the asymptotic behavior, as $t \rightarrow \infty$, of the occupation time field $\{T^x_t\}_{x \in Z^d}$, where $T^x_t = \int^t_0 \eta^\theta_s(x) ds$. Our main result is that, properly scaled and normalized, the occupation time field has a (weak) limit field as $t \rightarrow \infty$, whose covariance structure we compute explicitly. This field is Gaussian in dimensions $d \geq 2$. It is not Gaussian in dimension one, but has an "explicit" representation in terms of a system of coalescing Brownian motions. We also prove that $\lim_{t \rightarrow \infty} T^x_t/t = \theta$ a.s. for $d \geq 2$ (the result is false for $d = 1$). A striking feature of the behavior of the occupation time field is its elaborate dimension dependence.


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J. Theodore Cox. David Griffeath. "Occupation Time Limit Theorems for the Voter Model." Ann. Probab. 11 (4) 876 - 893, November, 1983.


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

zbMATH: 0527.60095
MathSciNet: MR714952
Digital Object Identifier: 10.1214/aop/1176993438

Primary: 60K35

Keywords: central limit theorems , Coalescing random walks , moments , Occupation times , semi-invariants , strong laws , Ursell functions , voter model

Rights: Copyright © 1983 Institute of Mathematical Statistics

Vol.11 • No. 4 • November, 1983
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