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August 1997 Multitype threshold growth: convergence to Poisson-Voronoi tessellations
Janko Gravner, David Griffeath
Ann. Appl. Probab. 7(3): 615-647 (August 1997). DOI: 10.1214/aoap/1034801246

Abstract

A Poisson-Voronoi tessellation (PVT) is a tiling of the Euclidean plane in which centers of individual tiles constitute a Poisson field and each tile comprises the locations that are closest to a given center with respect to a prescribed norm. Many spatial systems in which rare, randomly distributed centers compete for space should be well approximated by a PVT. Examples that we can handle rigorously include multitype threshold vote automata, in which $\kappa$ different camps compete for voters stationed on the two-dimensional lattice. According to the deterministic, discrete-time update rule, a voter changes affiliation only to that of a unique opposing camp having more than $\theta$ representatives in the voter's neighborhood. We establish a PVT limit for such dynamics started from completely random configurations, as the number of camps becomes large, so that the density of initial "pockets of consensus" tends to 0. Our methods combine nucleation analysis, Poisson approximation, and shape theory.

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Janko Gravner. David Griffeath. "Multitype threshold growth: convergence to Poisson-Voronoi tessellations." Ann. Appl. Probab. 7 (3) 615 - 647, August 1997. https://doi.org/10.1214/aoap/1034801246

Information

Published: August 1997
First available in Project Euclid: 16 October 2002

zbMATH: 0892.60096
MathSciNet: MR1459263
Digital Object Identifier: 10.1214/aoap/1034801246

Subjects:
Primary: 60K35
Secondary: 05B15

Keywords: Poisson convergence , random closed sets , threshold vote automata , Voronoi tessellation , weak convergence

Rights: Copyright © 1997 Institute of Mathematical Statistics

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Vol.7 • No. 3 • August 1997
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