The Annals of Probability

Occupation Time Limit Theorems for the Voter Model

J. Theodore Cox and David Griffeath

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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.

Article information

Ann. Probab. Volume 11, Number 4 (1983), 876-893.

First available in Project Euclid: 19 April 2007

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Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Voter model coalescing random walks occupation times central limit theorems strong laws moments semi-invariants Ursell functions


Cox, J. Theodore; Griffeath, David. Occupation Time Limit Theorems for the Voter Model. Ann. Probab. 11 (1983), no. 4, 876--893. doi:10.1214/aop/1176993438.

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