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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.79.

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Percolation in the hyperbolic plane
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by Itai Benjamini and Oded Schramm PDF
J. Amer. Math. Soc. 14 (2001), 487-507 Request permission

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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Additional 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