Percolation in the hyperbolic plane

Abstract

We study percolation in the hyperbolic plane H 2 \mathbb {H}^2 and on regular tilings in the hyperbolic plane. The processes discussed include Bernoulli site and bond percolation on planar hyperbolic graphs, invariant dependent percolations on such graphs, and Poisson-Voronoi-Bernoulli percolation. We prove the existence of three distinct nonempty phases for the Bernoulli processes. In the first phase, p ∈ ( 0 , p c ] p\in (0,p_c] , there are no unbounded clusters, but there is a unique infinite cluster for the dual process. In the second phase, p ∈ ( p c , p u ) p\in (p_c,p_u) , there are infinitely many unbounded clusters for the process and for the dual process. In the third phase, p ∈ [ p u , 1 ) p\in [p_u,1) , there is a unique unbounded cluster, and all the clusters of the dual process are bounded. We also study the dependence of p c p_c in the Poisson-Voronoi-Bernoulli percolation process on the intensity of the underlying Poisson process.