THE LOW-INTENSITY LIMIT OF BERNOULLI-VORONOI AND POISSON-VORONOI MEASURES

Abstract

On a graph G the Bernoulli-Voronoi measure is an edge percolation process obtained as a result of the following procedure; first select vertices by a Bernoulli site percolation,with intensity p, as nuclei for a Voronoi tessellation and then delete edges whose end vertices lie in different Voronoi tiles. A natural question which arises in this context is the existence of the weak limit as the intensity of the nuclei tends to zero. In this work, we study the existence and properties of this limit on d-regular trees and other infinite Cayley graphs. A related model in the continuous setting is the Poisson-Voronoi model, which can be defined on any metric space. In this setting, the nuclei of the Voronoi tiles are obtained by a Poisson process, with intensity lambda, and the random set we study is the boundary of the Voronoi tiles. We study various properties of the low-intensity limit, as lambda tends to zero, of this measure in the hyperbolic space.