Poisson-Voronoi percolation in the hyperbolic plane with small intensities

Abstract

We consider percolation on the Voronoi tessellation generated by a homogeneous Poisson point process on the hyperbolic plane. We show that the critical probability for the existence of an infinite cluster is asymptotically equal to $\pi \lambda/3$ as $\lambda\to0.$ This answers a question of Benjamini and Schramm.Comment: 70 pages, 11 figures, various corrections made compared to the last versio