## Percolation in the hyperbolic plane

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- by Itai Benjamini and Oded Schramm
- J. Amer. Math. Soc.
**14**(2001), 487-507 - DOI: https://doi.org/10.1090/S0894-0347-00-00362-3
- Published electronically: December 28, 2000
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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.## References

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## Bibliographic Information

**Itai Benjamini**- Affiliation: Department of Mathematics, Weizmann Institute of Science, Rehovot 76100, Israel
- MR Author ID: 311800
- Email: itai@wisdom.weizmann.ac.il
**Oded Schramm**- Affiliation: Department of Mathematics, Weizmann Institute of Science, Rehovot 76100, Israel
- Email: schramm@microsoft.com
- 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
- © Copyright 2000 American Mathematical Society
- Journal: J. Amer. Math. Soc.
**14**(2001), 487-507 - MSC (2000): Primary 82B43; Secondary 60K35, 60D05
- DOI: https://doi.org/10.1090/S0894-0347-00-00362-3
- MathSciNet review: 1815220