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Journal of the American Mathematical Society
Journal of the American Mathematical Society
ISSN 1088-6834(online) ISSN 0894-0347(print)

 

Percolation in the hyperbolic plane


Authors: Itai Benjamini and Oded Schramm
Journal: J. Amer. Math. Soc. 14 (2001), 487-507
MSC (2000): Primary 82B43; Secondary 60K35, 60D05
Published electronically: December 28, 2000
MathSciNet review: 1815220
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Abstract:

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.


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Additional Information

Itai Benjamini
Affiliation: Department of Mathematics, Weizmann Institute of Science, Rehovot 76100, Israel
Email: itai@wisdom.weizmann.ac.il

Oded Schramm
Affiliation: Department of Mathematics, Weizmann Institute of Science, Rehovot 76100, Israel
Email: schramm@microsoft.com

DOI: http://dx.doi.org/10.1090/S0894-0347-00-00362-3
PII: S 0894-0347(00)00362-3
Keywords: Poisson process, Voronoi, isoperimetric constant, nonamenable, planarity, Euler formula, Gauss-Bonnet theorem
Received by editor(s): January 18, 2000
Received by editor(s) in revised form: November 9, 2000
Published electronically: December 28, 2000
Additional Notes: The second author’s research was partially supported by the Sam and Ayala Zacks Professorial Chair at the Weizmann Institute
Article copyright: © Copyright 2000 American Mathematical Society