Percolation in the Hyperbolic Plane
نویسنده
چکیده
Following is a study of percolation in the hyperbolic plane H 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.
منابع مشابه
Percolation in the Hyperbolic Plane
The purpose of this paper is to study percolation in the hyperbolic plane and in transitive planar graphs that are quasi-isometric to the hyperbolic plane. There are several sources available which the reader may consult for background on percolation on Z [Gri89] and R [MR96] and for background on percolation on more general graphs [BS96], [Lyo00], [BS99]. For this reason, we will be quite brie...
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