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October, 1989 Coalescing Random Walks and Voter Model Consensus Times on the Torus in $\mathbb{Z}^d$
J. T. Cox
Ann. Probab. 17(4): 1333-1366 (October, 1989). DOI: 10.1214/aop/1176991158

Abstract

Let $\eta_t$ be the basic voter model on $\mathbb{Z}^d$ and let $\eta^{(N)}_t$ be the voter model on $\Lambda(N)$, the torus of side $N$ in $\mathbb{Z}^d$. Unlike $\eta_t, \eta^{(N)}_t$ (for fixed $N$) gets trapped with probability 1 as $t \rightarrow\infty$ at all 0's or all 1's. We examine the asymptotic growth of these trapping or consensus times $\tau^{(N)}$ as $N \rightarrow\infty$. To do this we obtain limit theorems for coalescing random walk systems on the torus $\Lambda(N)$, including a new hitting time limit theorem for (noncoalescing) random walk on the torus.

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J. T. Cox. "Coalescing Random Walks and Voter Model Consensus Times on the Torus in $\mathbb{Z}^d$." Ann. Probab. 17 (4) 1333 - 1366, October, 1989. https://doi.org/10.1214/aop/1176991158

Information

Published: October, 1989
First available in Project Euclid: 19 April 2007

zbMATH: 0685.60100
MathSciNet: MR1048930
Digital Object Identifier: 10.1214/aop/1176991158

Subjects:
Primary: 60K35

Keywords: finite particle systems , infinite particle systems , Random walks , voter model

Rights: Copyright © 1989 Institute of Mathematical Statistics

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Vol.17 • No. 4 • October, 1989
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