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

DOI:
https://doi.org/10.1090/S0894-0347-00-00362-3

Published electronically:
December 28, 2000

MathSciNet review:
1815220

Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 Poisson-Voronoi-Bernoulli 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 Poisson-Voronoi-Bernoulli percolation process on the intensity of the underlying Poisson process.

**[Bab97]**L. Babai,*The growth rate of vertex-transitive planar graphs*, in Proceedings of the Eighth Annual ACM-SIAM Symposium on Discrete Algorithms (New Orleans, LA, 1997), ACM, New York, 1997, pp. 564-573. MR**97k:68011****[BK89]**R. M. Burton and M. Keane,*Density and uniqueness in percolation*, Comm. Math. Phys.**121**(1989), no. 3, 501-505. MR**90g:60090****[BK91]**-,*Topological and metric properties of infinite clusters in stationary two-dimensional site percolation*, Israel J. Math.**76**(1991), no. 3, 299-316. MR**93i:60182****[BLPS99a]**I. Benjamini, R. Lyons, Y. Peres, and O. Schramm,*Group-invariant percolation on graphs*, Geom. Funct. Anal.**9**(1999), no. 1, 29-66. MR**99m:60149****[BLPS99b]**-,*Critical percolation on any nonamenable group has no infinite clusters*, Ann. Probab.**27**(1999), no. 3, 1347-1356. 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. 56-84. Papers from the workshop held in Cortona, 1997.**[BS96]**I. Benjamini and O. Schramm,*Percolation beyond , many questions and a few answers*, Electron. Comm. Probab.**1**(1996), no. 8, 71-82 (electronic). MR**97j:60179****[BS99]**-,*Recent progress on percolation beyond*, http://www.wisdom.weizmann.ac.il/ schramm/papers/pyond-rep/, 1999.**[BSt90]**A. F. Beardon and K. Stephenson,*The uniformization theorem for circle packings*, Indiana Univ. Math. J.**39**(1990), no. 4, 1383-1425. 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. 59-115. MR**99c:57036****[DS84]**P. G. Doyle and J. L. Snell,*Random walks and electric networks*, Mathematical Association of America, Washington, DC, 1984. MR**89a:94023****[GN90]**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. 167-190. MR**92a:60207****[Gri89]**G. Grimmett,*Percolation*, Springer-Verlag, New York, 1989. MR**90j:60109****[Häg97]**O. Häggström,*Infinite clusters in dependent automorphism invariant percolation on trees*, Ann. Probab.**25**(1997), no. 3, 1423-1436. MR**98f:60207****[HP99]**O. Häggström and Y. Peres,*Monotonicity of uniqueness for percolation on Cayley graphs: all infinite clusters are born simultaneously*, Probab. Theory Related Fields**113**(1999), no. 2, 273-285. MR**99k:60253****[HS95]**Z.-X. He and O. Schramm,*Hyperbolic and parabolic packings*, Discrete Comput. Geom.**14**(1995), no. 2, 123-149. MR**96h:52017****[Imr75]**W. Imrich,*On Whitney's theorem on the unique embeddability of 3-connected planar graphs*, in Recent Advances in Graph Theory (Proc. Second Czechoslovak Sympos., Prague, 1974), Academia, Prague, 1975, pp. 303-306. MR**52:5462****[Lal98]**S. P. Lalley,*Percolation on Fuchsian groups*, Ann. Inst. H. Poincaré Probab. Statist.**34**(1998), no. 2, 151-177. MR**99g:60190****[Lyo00]**R. Lyons,*Phase transitions on nonamenable graphs*, J. Math. Phys.**41**(2000), no. 3, 1099-1126. Probabilistic techniques in equilibrium and nonequilibrium statistical physics. CMP**2000:13****[Lyo01]**-,*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/.**[Mad70]**W. Mader,*Über den Zusammenhang symmetrischer Graphen*, Arch. Math. (Basel)**21**(1970), 331-336. MR**44:6534****[MR96]**R. Meester and R. Roy,*Continuum percolation*, Cambridge University Press, Cambridge, 1996. MR**98d:60193****[PSN00]**I. Pak and T. Smirnova-Nagnibeda,*On non-uniqueness of percolation on non-amenable Cayley graphs*, C. R. Acad. Sci. Paris Sér. I Math.**330**(2000), no. 6, 495-500. MR**2000m:60116****[Per00]**Y. Peres,*Percolation on nonamenable products at the uniqueness threshold*, Ann. Inst. H. Poincaré Probab. Statist.**36**(2000), no. 3, 395-406. 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. 53-67. CMP**99:16****[Sch99b]**-,*Stability of infinite clusters in supercritical percolation*, Probab. Theory Related Fields**113**(1999), no. 2, 287-300. 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. 9-19. MR**98c:11064****[Wat70]**M. E. Watkins,*Connectivity of transitive graphs*, J. Combinatorial Theory**8**(1970), 23-29. MR**42:1707****[Zva96]**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:
https://doi.org/10.1090/S0894-0347-00-00362-3

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