Percolation in the hyperbolic plane

Benjamini, Itai¹ ; Schramm, Oded¹

¹ Department of Mathematics, Weizmann Institute of Science, Rehovot 76100, Israel

Journal of the American Mathematical Society, Tome 14 (2001) no. 2, pp. 487-507

Résumé

We study percolation in the hyperbolic plane $\mathbb {H}^2$ 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, $p\in (0,p_c]$, there are no unbounded clusters, but there is a unique infinite cluster for the dual process. In the second phase, $p\in (p_c,p_u)$, there are infinitely many unbounded clusters for the process and for the dual process. In the third phase, $p\in [p_u,1)$, there is a unique unbounded cluster, and all the clusters of the dual process are bounded. We also study the dependence of $p_c$ in the Poisson-Voronoi-Bernoulli percolation process on the intensity of the underlying Poisson process.

Détail

DOI : 10.1090/S0894-0347-00-00362-3

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