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January 1999 The Complete Convergence Theorem for Coexistent Threshold Voter Models
Shirin J. Handjani
Ann. Probab. 27(1): 226-245 (January 1999). DOI: 10.1214/aop/1022677260

Abstract

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.

Citation

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Shirin J. Handjani. "The Complete Convergence Theorem for Coexistent Threshold Voter Models." Ann. Probab. 27 (1) 226 - 245, January 1999. https://doi.org/10.1214/aop/1022677260

Information

Published: January 1999
First available in Project Euclid: 29 May 2002

zbMATH: 0974.60093
MathSciNet: MR1681118
Digital Object Identifier: 10.1214/aop/1022677260

Subjects:
Primary: 60K35

Rights: Copyright © 1999 Institute of Mathematical Statistics

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Vol.27 • No. 1 • January 1999
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