Percolation in the Hyperbolic Plane

Abstract

Following is a study of percolation in the hyperbolic plane H² 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, pc], there are no unbounded clusters, but there is a unique infinite cluster for the dual process. In the second phase, p ∈ (pc, pu), there are infinitely many unbounded clusters for the process and for the dual process. In the third phase, p∈[pu, 1), there is a unique unbounded cluster, and all the clusters of the dual process are bounded. We also study the dependence of pc in the Poisson-Voronoi-Bernoulli percolation process on the intensity of the underlying Poisson process.