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Volumn 14, Issue 2, 2001, Pages 487-507

Percolation in the hyperbolic plane

Author keywords

Euler formula; Gauss Bonnet theorem; Isoperimetric constant; Nonamenable; Planarity; Poisson process; Voronoi

Indexed keywords


EID: 0035585114     PISSN: 08940347     EISSN: None     Source Type: Journal    
DOI: 10.1090/S0894-0347-00-00362-3     Document Type: Article
Times cited : (128)

References (55)
  • 2
    • 0030834634 scopus 로고    scopus 로고
    • 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
    • MR , vol.97 K , pp. 68011
  • 3
    • 0000393829 scopus 로고    scopus 로고
    • Density and uniqueness in percolation
    • R. M. Burton and M. Keane, Density and uniqueness in percolation, Comm. Math. Phys. 121 (1989), no. 3, 501-505. MR 90g:60090
    • (1989) Comm. Math. Phys. , vol.121 , Issue.3 , pp. 501-505
    • Burton, R.M.1    Keane, M.2
  • 4
    • 0000393829 scopus 로고    scopus 로고
    • R. M. Burton and M. Keane, Density and uniqueness in percolation, Comm. Math. Phys. 121 (1989), no. 3, 501-505. MR 90g:60090
    • MR , vol.90 G , pp. 60090
  • 5
    • 51249175677 scopus 로고    scopus 로고
    • Topological and metric properties of infinite clusters in stationary two-dimensional site percolation
    • _, 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
    • (1991) Israel J. Math. , vol.76 , Issue.3 , pp. 299-316
  • 6
    • 51249175677 scopus 로고    scopus 로고
    • _, 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
    • MR , vol.93 I , pp. 60182
  • 8
    • 0033447464 scopus 로고    scopus 로고
    • 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
    • MR , vol.99 M , pp. 60149
  • 9
    • 0033164323 scopus 로고    scopus 로고
    • Critical percolation on any nonamenable group has no infinite clusters
    • _, Critical percolation on any nonamenable group has no infinite clusters, Ann. Probab. 27 (1999), no. 3, 1347-1356. MR 2000k:60197
    • (1999) Ann. Probab. , vol.27 , Issue.3 , pp. 1347-1356
  • 10
    • 0033164323 scopus 로고    scopus 로고
    • _, Critical percolation on any nonamenable group has no infinite clusters, Ann. Probab. 27 (1999), no. 3, 1347-1356. MR 2000k:60197
    • MR , vol.2000 K , pp. 60197
  • 11
    • 0347276507 scopus 로고    scopus 로고
    • Uniform spanning forests
    • to appear
    • _, Uniform spanning forests, Ann. Probab. (2000), to appear.
    • (2000) Ann. Probab.
  • 12
    • 0001966997 scopus 로고    scopus 로고
    • Percolation perturbations in potential theory and random walks
    • M. Picardello and W. Woess, editors, Cambridge University Press, Cambridge, Papers from the workshop held in Cortona, 1997
    • 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.
    • (1999) Random Walks and Discrete Potential Theory, Sympos. Math. , pp. 56-84
    • Benjamini, I.1    Lyons, R.2    Schramm, O.3
  • 13
    • 0001883932 scopus 로고    scopus 로고
    • d, many questions and a few answers
    • electronic
    • d, many questions and a few answers, Electron. Comm. Probab. 1 (1996), no. 8, 71-82 (electronic). MR 97j:60179
    • (1996) Electron. Comm. Probab. , vol.1 , Issue.8 , pp. 71-82
    • Benjamini, I.1    Schramm, O.2
  • 14
    • 0001883932 scopus 로고    scopus 로고
    • d, many questions and a few answers, Electron. Comm. Probab. 1 (1996), no. 8, 71-82 (electronic). MR 97j:60179
    • MR , vol.97 J , pp. 60179
  • 15
    • 0003964674 scopus 로고    scopus 로고
    • d, http://wwu.wisdom.weizmann.ac. il/~schramm/papers/pyond-rep/, 1999.
    • (1999) d
  • 16
    • 0001313129 scopus 로고
    • The uniformization theorem for circle packings
    • 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
    • (1990) Indiana Univ. Math. J. , vol.39 , Issue.4 , pp. 1383-1425
    • Beardon, A.F.1    Stephenson, K.2
  • 17
    • 0346646495 scopus 로고    scopus 로고
    • 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
    • MR , vol.92 B , pp. 52038
  • 18
    • 0003064406 scopus 로고    scopus 로고
    • Hyperbolic geometry
    • Cambridge University Press, Cambridge
    • 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
    • (1997) Flavors of Geometry , pp. 59-115
    • Cannon, J.W.1    Floyd, W.J.2    Kenyon, R.3    Parry, W.R.4
  • 19
    • 0347276502 scopus 로고    scopus 로고
    • 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
    • MR , vol.99 C , pp. 57036
  • 21
    • 0040267016 scopus 로고    scopus 로고
    • P. G. Doyle and J. L. Snell, Random walks and electric networks, Mathematical Association of America, Washington, DC, 1984. MR 89a:94023
    • MR , vol.89 A , pp. 94023
  • 22
    • 0000042426 scopus 로고
    • Percolation in ∞ + 1 dimensions
    • G. R. Grimmett and D. J. A. Welsh, editors, Oxford University Press, New York
    • G. R. Grimmett and C. M. Newman, Percolation in ∞ + 1 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
    • (1990) Disorder in Physical Systems , pp. 167-190
    • Grimmett, G.R.1    Newman, C.M.2
  • 23
    • 0346015368 scopus 로고    scopus 로고
    • G. R. Grimmett and C. M. Newman, Percolation in ∞ + 1 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
    • MR , vol.92 A , pp. 60207
  • 24
    • 0004246476 scopus 로고
    • Springer-Verlag, New York
    • G. Grimmett, Percolation, Springer-Verlag, New York, 1989. MR 90j:60109
    • (1989) Percolation
    • Grimmett, G.1
  • 25
    • 0346646499 scopus 로고    scopus 로고
    • G. Grimmett, Percolation, Springer-Verlag, New York, 1989. MR 90j:60109
    • MR , vol.90 J , pp. 60109
  • 26
    • 0040453578 scopus 로고    scopus 로고
    • Infinite clusters in dependent automorphism invariant percolation on trees
    • O. Häggström, Infinite clusters in dependent automorphism invariant percolation on trees, Ann. Probab. 25 (1997), no. 3, 1423-1436. MR 98f:60207
    • (1997) Ann. Probab. , vol.25 , Issue.3 , pp. 1423-1436
    • Häggström, O.1
  • 27
    • 0040453578 scopus 로고    scopus 로고
    • O. Häggström, Infinite clusters in dependent automorphism invariant percolation on trees, Ann. Probab. 25 (1997), no. 3, 1423-1436. MR 98f:60207
    • MR , vol.98 F , pp. 60207
  • 28
    • 0038934720 scopus 로고    scopus 로고
    • Monotonicity of uniqueness for percolation on Cayley graphs: All infinite clusters are born simultaneously
    • 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
    • (1999) Probab. Theory Related Fields , vol.113 , Issue.2 , pp. 273-285
    • Häggström, O.1    Peres, Y.2
  • 29
    • 0038934720 scopus 로고    scopus 로고
    • 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
    • MR , vol.99 K , pp. 60253
  • 30
    • 0009305212 scopus 로고    scopus 로고
    • Hyperbolic and parabolic packings
    • Z.-X. He and O. Schramm, Hyperbolic and parabolic packings, Discrete Comput. Geom. 14 (1995), no. 2, 123-149. MR 96h:52017
    • (1995) Discrete Comput. Geom. , vol.14 , Issue.2 , pp. 123-149
    • He, Z.-X.1    Schramm, O.2
  • 31
    • 0009305212 scopus 로고    scopus 로고
    • Z.-X. He and O. Schramm, Hyperbolic and parabolic packings, Discrete Comput. Geom. 14 (1995), no. 2, 123-149. MR 96h:52017
    • MR , vol.96 H , pp. 52017
  • 32
    • 0000238045 scopus 로고
    • On Whitney's theorem on the unique embeddability of 3-connected planar graphs
    • Recent Advances in Graph Theory Academia, Prague
    • 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
    • (1974) Proc. Second Czechoslovak Sympos., Prague , pp. 303-306
    • Imrich, W.1
  • 33
    • 0346015362 scopus 로고    scopus 로고
    • 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
    • MR , vol.52 , pp. 5462
  • 34
  • 35
    • 21844454507 scopus 로고    scopus 로고
    • S. P. Lalley, Percolation on Fuchsian groups, Ann. Inst. H. Poincaré Probab. Statist. 34 (1998), no. 2, 151-177. MR 99g:60190
    • MR , vol.99 G , pp. 60190
  • 36
    • 0034345423 scopus 로고    scopus 로고
    • Phase transitions on nonamenable graphs
    • Probabilistic techniques in equilibrium and nonequilibrium statistical physics
    • 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
    • (2000) J. Math. Phys. , vol.41 , Issue.3 , pp. 1099-1126
    • Lyons, R.1
  • 37
    • 0034345423 scopus 로고    scopus 로고
    • 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
    • CMP , vol.2000 , pp. 13
  • 38
    • 0003458515 scopus 로고    scopus 로고
    • Cambridge University Press, Written with the assistance of Y. Peres, in preparation. Current version available at
    • _, 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/.
    • (2001) Probability on Trees and Networks
  • 39
    • 0040889325 scopus 로고    scopus 로고
    • Über den zusammenhang symmetrischer graphen
    • W. Mader, Über den Zusammenhang symmetrischer Graphen, Arch. Math. (Basel) 21 (1970), 331-336. MR 44:6534
    • (1970) Arch. Math. (Basel) , vol.21 , pp. 331-336
    • Mader, W.1
  • 40
    • 0040889325 scopus 로고    scopus 로고
    • W. Mader, Über den Zusammenhang symmetrischer Graphen, Arch. Math. (Basel) 21 (1970), 331-336. MR 44:6534
    • MR , vol.44 , pp. 6534
  • 41
    • 0004111265 scopus 로고    scopus 로고
    • Cambridge University Press, Cambridge
    • R. Meester and R. Roy, Continuum percolation, Cambridge University Press, Cambridge, 1996. MR 98d:60193
    • (1996) Continuum Percolation
    • Meester, R.1    Roy, R.2
  • 42
    • 0346015361 scopus 로고    scopus 로고
    • R. Meester and R. Roy, Continuum percolation, Cambridge University Press, Cambridge, 1996. MR 98d:60193
    • MR , vol.98 D , pp. 60193
  • 43
    • 0034652842 scopus 로고    scopus 로고
    • On non-uniqueness of percolation on nonamenable Cayley graphs
    • I. Pak and T. Smirnova-Nagnibeda, On non-uniqueness of percolation on nonamenable Cayley graphs, C. R. Acad. Sci. Paris Sér. I Math. 330 (2000), no. 6, 495-500. MR 2000m:60116
    • (2000) C. R. Acad. Sci. Paris Sér. I Math. , vol.330 , Issue.6 , pp. 495-500
    • Pak, I.1    Smirnova-Nagnibeda, T.2
  • 44
    • 0034652842 scopus 로고    scopus 로고
    • I. Pak and T. Smirnova-Nagnibeda, On non-uniqueness of percolation on nonamenable Cayley graphs, C. R. Acad. Sci. Paris Sér. I Math. 330 (2000), no. 6, 495-500. MR 2000m:60116
    • MR , vol.2000 M , pp. 60116
  • 45
    • 0042686172 scopus 로고    scopus 로고
    • Percolation on nonamenable products at the uniqueness threshold
    • 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
    • (2000) Ann. Inst. H. Poincaré Probab. Statist. , vol.36 , Issue.3 , pp. 395-406
    • Peres, Y.1
  • 46
    • 0042686172 scopus 로고    scopus 로고
    • 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
    • CMP , vol.2000 , pp. 15
  • 47
    • 0001791767 scopus 로고    scopus 로고
    • Percolation in ∞ + 1 dimensions at the uniqueness threshold
    • M. Bramson and R. Durrett, editors, Birkhäuser, Boston
    • R. H. Schonmann, Percolation in ∞ + 1 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
    • (1999) Perplexing Probability Problems: Papers in Honor of Harry Kesten , pp. 53-67
    • Schonmann, R.H.1
  • 48
    • 0347275630 scopus 로고    scopus 로고
    • R. H. Schonmann, Percolation in ∞ + 1 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
    • CMP , vol.99 , pp. 16
  • 49
    • 0033077109 scopus 로고    scopus 로고
    • Stability of infinite clusters in supercritical percolation
    • _, Stability of infinite clusters in supercritical percolation, Probab. Theory Related Fields 113 (1999), no. 2, 287-300. MR 99k:60252
    • (1999) Probab. Theory Related Fields , vol.113 , Issue.2 , pp. 287-300
  • 50
    • 0033077109 scopus 로고    scopus 로고
    • _, Stability of infinite clusters in supercritical percolation, Probab. Theory Related Fields 113 (1999), no. 2, 287-300. MR 99k:60252
    • MR , vol.99 K , pp. 60252
  • 51
    • 0346014458 scopus 로고
    • Extremal Riemann surfaces with a large number of systoles
    • San Francisco, CA, Amer. Math. Soc., Providence, RI
    • 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
    • (1995) Extremal Riemann Surfaces , pp. 9-19
    • Schmutz Schaller, P.1
  • 52
    • 0347275629 scopus 로고    scopus 로고
    • 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
    • MR , vol.98 C , pp. 11064
  • 53
    • 0002443025 scopus 로고    scopus 로고
    • Connectivity of transitive graphs
    • M. E. Watkins, Connectivity of transitive graphs, J. Combinatorial Theory 8 (1970), 23-29. MR 42:1707
    • (1970) J. Combinatorial Theory , vol.8 , pp. 23-29
    • Watkins, M.E.1
  • 54
    • 0002443025 scopus 로고    scopus 로고
    • M. E. Watkins, Connectivity of transitive graphs, J. Combinatorial Theory 8 (1970), 23-29. MR 42:1707
    • MR , vol.42 , pp. 1707


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