Percolation in the hyperbolic plane
Authors:
Itai Benjamini and Oded Schramm
Journal:
J. Amer. Math. Soc. 14 (2001), 487507
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 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 PoissonVoronoiBernoulli percolation. We prove the existence of three distinct nonempty phases for the Bernoulli processes. In the first phase, , there are no unbounded clusters, but there is a unique infinite cluster for the dual process. In the second phase, , there are infinitely many unbounded clusters for the process and for the dual process. In the third phase, , there is a unique unbounded cluster, and all the clusters of the dual process are bounded. We also study the dependence of in the PoissonVoronoiBernoulli percolation process on the intensity of the underlying Poisson process.
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 I. Benjamini, R. Lyons, Y. Peres, and O. Schramm, Groupinvariant percolation on graphs, Geom. Funct. Anal. 9 (1999), no. 1, 2966. MR 99m:60149
 [BLPS99b]
 , Critical percolation on any nonamenable group has no infinite clusters, Ann. Probab. 27 (1999), no. 3, 13471356. MR 2000k:60197
 [BLPS00]
 , Uniform spanning forests, Ann. Probab. (2000), to appear.
 [BLS99]
 I. Benjamini, R. Lyons, and O. Schramm, Percolation perturbations in potential theory and random walks, in M. Picardello and W. Woess, editors, Random Walks and Discrete Potential Theory, Sympos. Math., Cambridge University Press, Cambridge, 1999, pp. 5684. Papers from the workshop held in Cortona, 1997.
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 [BS99]
 , Recent progress on percolation beyond , http://www.wisdom.weizmann.ac.il/ schramm/papers/pyondrep/, 1999.
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 A. F. Beardon and K. Stephenson, The uniformization theorem for circle packings, Indiana Univ. Math. J. 39 (1990), no. 4, 13831425. MR 92b:52038
 [CFKP97]
 J. W. Cannon, W. J. Floyd, R. Kenyon, and W. R. Parry, Hyperbolic geometry, in Flavors of Geometry, Cambridge University Press, Cambridge, 1997, pp. 59115. MR 99c:57036
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 G. R. Grimmett and C. M. Newman, Percolation in dimensions, in G. R. Grimmett and D. J. A. Welsh, editors, Disorder in Physical Systems, Oxford University Press, New York, 1990, pp. 167190. MR 92a:60207
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 S. P. Lalley, Percolation on Fuchsian groups, Ann. Inst. H. Poincaré Probab. Statist. 34 (1998), no. 2, 151177. MR 99g:60190
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 R. Lyons, Phase transitions on nonamenable graphs, J. Math. Phys. 41 (2000), no. 3, 10991126. Probabilistic techniques in equilibrium and nonequilibrium statistical physics. CMP 2000:13
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 , Probability on trees and networks, Cambridge University Press, 2001. Written with the assistance of Y. Peres, in preparation. Current version available at http://php.indiana.edu/ rdlyons/.
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 R. Meester and R. Roy, Continuum percolation, Cambridge University Press, Cambridge, 1996. MR 98d:60193
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 I. Pak and T. SmirnovaNagnibeda, On nonuniqueness of percolation on nonamenable Cayley graphs, C. R. Acad. Sci. Paris Sér. I Math. 330 (2000), no. 6, 495500. MR 2000m:60116
 [Per00]
 Y. Peres, Percolation on nonamenable products at the uniqueness threshold, Ann. Inst. H. Poincaré Probab. Statist. 36 (2000), no. 3, 395406. CMP 2000:15
 [Sch99a]
 R. H. Schonmann, Percolation in dimensions at the uniqueness threshold, in M. Bramson and R. Durrett, editors, Perplexing Probability Problems: Papers in Honor of Harry Kesten, Birkhäuser, Boston, 1999, pp. 5367. CMP 99:16
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 , Stability of infinite clusters in supercritical percolation, Probab. Theory Related Fields 113 (1999), no. 2, 287300. MR 99k:60252
 [SS97]
 P. Schmutz Schaller, Extremal Riemann surfaces with a large number of systoles, in Extremal Riemann Surfaces (San Francisco, CA, 1995), Amer. Math. Soc., Providence, RI, 1997, pp. 919. MR 98c:11064
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 M. E. Watkins, Connectivity of transitive graphs, J. Combinatorial Theory 8 (1970), 2329. MR 42:1707
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 A. Zvavitch, The critical probability for Voronoi percolation, MSc. thesis, Weizmann Institute of Science, 1996.
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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/S0894034700003623
PII:
S 08940347(00)003623
Keywords:
Poisson process,
Voronoi,
isoperimetric constant,
nonamenable,
planarity,
Euler formula,
GaussBonnet 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
