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