Percolation in the hyperbolic plane

Abstract

We study percolation in the hyperbolic planeH2\mathbb {H}^2and 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,pc]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∈(pc,pu)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∈[pu,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 ofpcp_cin the Poisson-Voronoi-Bernoulli percolation process on the intensity of the underlying Poisson process.