The Annals of Probability

The Complete Convergence Theorem for Coexistent Threshold Voter Models

Shirin J. Handjani

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We consider the $d$-dimensional threshold voter model. It is known that, except in the one-dimensional nearest-neighbor case, coexistence occurs (nontrivial invariant measures exist). In fact, there is a nontrivial limit $\eta_\infty^{1/2}$ obtained by starting from the product measure with density 1/2. We show that in these coexistent cases,

\eta_t \Rightarrow \alpha\delta_0 + \beta\delta_1 + (1 - \alpha - \beta)\eta^{1/2}_\infty \quad\text{as $t \to \infty$},

where $\alpha=P(\tau_0<\infty), \beta=P(\tau_1<\infty), \tau_0$ and $\tau_1$ are the first hitting times of the all-zero and all-one configurations, respectively, and $\Rightarrow$ denotes weak convergence.

Article information

Ann. Probab., Volume 27, Number 1 (1999), 226-245.

First available in Project Euclid: 29 May 2002

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Zentralblatt MATH identifier

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

Spin systems coexistence voter models complete convergence


Handjani, Shirin J. The Complete Convergence Theorem for Coexistent Threshold Voter Models. Ann. Probab. 27 (1999), no. 1, 226--245. doi:10.1214/aop/1022677260.

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