The Annals of Probability

Super-Brownian Limits of Voter Model Clusters

Maury Bramson, J.Theodore Cox, and Jean-François Le Gall

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The voter model is one of the standard interacting particle systems. Two related problems for this process are to analyze its behavior, after large times $t$, for the sets of sites (1) sharing the same opinion as the site 0, and (2) having the opinion that was originally at 0. Results on the sizes of these sets were given by Sawyer (1979)and Bramson and Griffeath (1980). Here, we investigate the spatial structure of these sets in $d \geq 2$, which we show converge to quantities associated with super-Brownian motion, after suitable normalization. The main theorem from Cox, Durrett and Perkins (2000) serves as an important tool for these results.

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Ann. Probab., Volume 29, Number 3 (2001), 1001-1032.

First available in Project Euclid: 5 March 2002

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Zentralblatt MATH identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 60G57: Random measures
Secondary: 60F05: Central limit and other weak theorems 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)

Voter model super-Brownian motion coalescing random walk


Bramson, Maury; Cox, J.Theodore; Le Gall, Jean-François. Super-Brownian Limits of Voter Model Clusters. Ann. Probab. 29 (2001), no. 3, 1001--1032. doi:10.1214/aop/1015345593.

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