The Annals of Probability
- Ann. Probab.
- Volume 29, Number 3 (2001), 1001-1032.
Super-Brownian Limits of Voter Model Clusters
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.
Ann. Probab., Volume 29, Number 3 (2001), 1001-1032.
First available in Project Euclid: 5 March 2002
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
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.)
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. https://projecteuclid.org/euclid.aop/1015345593