Hostname: page-component-6766d58669-76mfw Total loading time: 0 Render date: 2026-05-18T18:41:18.235Z Has data issue: false hasContentIssue false
  • English
  • Français

Percolation of finite clusters and infinite surfaces

Published online by Cambridge University Press:  15 January 2014

GEOFFREY R. GRIMMETT ,
ALEXANDER E. HOLROYD  and
GADY KOZMA
GEOFFREY R. GRIMMETT
Affiliation:
Statistical Laboratory, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WB. e-mail: g.r.grimmett@statslab.cam.ac.uk, URL: http://www.statslab.cam.ac.uk/~grg/
ALEXANDER E. HOLROYD
Affiliation:
Microsoft Research, 1 Microsoft Way, Redmond, WA 98052, U.S.A. e-mail: holroyd@microsoft.com, URL: http://research.microsoft.com/~holroyd/
GADY KOZMA
Affiliation:
Department of Mathematics, Weizmann Institute of Science, POB 26, Rehovot 76100, Israel. e-mail: gady.kozma@weizmann.ac.il, URL: http://www.wisdom.weizmann.ac.il/~gadyk/
Get access Rights & Permissions [Opens in a new window]

Abstract

Two related issues are explored for bond percolation on ${\mathbb{Z}^d$ (with d ≥ 3) and its dual plaquette process. Firstly, for what values of the parameter p does the complement of the infinite open cluster possess an infinite component? The corresponding critical point pfin satisfies pfin ≥ pc, and strict inequality is proved when either d is sufficiently large, or d ≥ 7 and the model is sufficiently spread out. It is not known whether d ≥ 3 suffices. Secondly, for what p does there exist an infinite dual surface of plaquettes? The associated critical point psurf satisfies psurf ≥ pfin.

Information

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 156 , Issue 2 , March 2014 , pp. 263 - 279
Copyright
Copyright © Cambridge Philosophical Society 2014 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

REFERENCES

[1]Ahlberg, D., Duminil–Copin, H., Kozma, G. and Sidoravicius, V. Seven-dimensional forest fires (2013), available at: arXiv:1302.6872.Google Scholar
[2]Aizenman, M. and Barsky, D. J.Sharpness of the phase transition in percolation models. Commun. Math. Phys. 108 (1987), 489–526.Google Scholar
[3]Aizenman, M., Chayes, J. T., Chayes, L., Fröhlich, J. and Russo, L.On a sharp transition from area law to perimeter law in a system of random surfaces. Commun. Math. Phys. 92 (1983), 19–69.Google Scholar
[4]Ballasteros, H. G., Fernández, A., Martín-Major, V., Sudupe, M. Muñoz, Parisi, G. and Ruiz-Lorenzo, J. J.Measures of critical exponents in the four dimensional site percolation. Phys. Lett. B 400 (1997), 346–351.Google Scholar
[5]Benjamini, I., Lyons, R., Peres, Y. and Schramm, O.Group-invariant percolation on graphs. Geom. Funct. Anal. 9 (1999), 29–66.Google Scholar
[6]Bollobás, B. and Leader, I.Compressions and isoperimetric inequalities. J. Combin. Theory Ser. A 56 (1991), 47–62.CrossRefGoogle Scholar
[7]Burton, R. M. and Keane, M.Density and uniqueness in percolation. Commun. Math. Phys. 121 (1989), 501–505.Google Scholar
[8]Campanino, M. and Russo, L.An upper bound on the critical percolation probability for the three-dimensional cubic lattice. Ann. Probab. 13 (1985), 478–491.Google Scholar
[9]Deuschel, J.-D. and Pisztora, Á.Surface order large deviations for high-density percolation. Probab. Theory Related Fields 104 (1996), 467–482.Google Scholar
[10]Dirr, N., Dondl, P. W., Grimmett, G. R., Holroyd, A. E. and Scheutzow, M.Lipschitz percolation. Electron. Commun. Probab. 15 (2010), 14–21.Google Scholar
[11]Grimmett, G. R.Percolation, 2nd ed. (Springer, Berlin, 1999).Google Scholar
[12]Grimmett, G. R.The Random-Cluster Model (Springer, Berlin, 2006).Google Scholar
[13]Grimmett, G. R. and Holroyd, A. E.Entanglement in percolation. Proc. London Math. Soc. 81 (2000), 485–512.Google Scholar
[14]Grimmett, G. R. and Holroyd, A. E.Plaquettes, spheres, and entanglement. Electron. J. Probab. 15 (2010), 1415–1428.Google Scholar
[15]Grimmett, G. R. and Janson, S.Branching processes, and random-cluster measures on trees. J. European Math. Soc. 7 (2005), 253–281.Google Scholar
[16]Hughes, B. D.Random Walks and Random Environments. Volume 2: Random Environments (Clarendon Press, Oxford, 1996).CrossRefGoogle Scholar
[17]Jonasson, J.The random cluster model on a general graph and a phase transition characterization of nonamenability. Stoch. Proc. Appl. 79 (1999), 335–354.Google Scholar
[18]Kesten, H.Percolation Theory for Mathematicians (Birkhäuser, Boston, Mass., 1982).Google Scholar
[19]Kesten, H.Scaling relations for 2D-percolation. Commun. Math. Phys. 109 (1987), 109–156.Google Scholar
[20]Kozma, G. and Nachmias, A.Arm exponents in high dimensional percolation. J. Amer. Math. Soc. 24 (2011), 375–409.Google Scholar
[21]Laanait, L., Messager, A., Miracle-Solé, S., Ruiz, J. and Shlosman, S.Interfaces in the Potts model I: Pirogov–Sinai theory of the Fortuin–Kasteleyn representation. Commun. Math. Phys. 140 (1991), 81–91.Google Scholar
[22]Lawler, G. F., Schramm, O. and Werner, W.One-arm exponent for 2D critical percolation. Electron. J. Probab. 7 (2002), 1–13.Google Scholar
[23]Liggett, T. M., Schonmann, R. H. and Stacey, A. M.Domination by product measures. Ann. Probab. 25 (1997), 71–95.Google Scholar
[24]Lorenz, C. D. and Ziff, R. M.Precise determination of the bond percolation thresholds and finite-size scaling corrections for the sc, fcc and bcc lattices. Phys. Rev. E 57 (1998), 230–236.Google Scholar
[25]Lyons, R. with Peres, Y. Probability on trees and networks. 2012 In preparation, available at: iu.edu~rdlyons.Google Scholar
[26]Menshikov, M. V.Coincidence of critical points in percolation problems. Dokl. Akad. Nauk SSSR 288 (1986), 1308–1311.Google Scholar
[27]Menshikov, M. V., Molchanov, S. A. and Sidorenko, A. F.Percolation theory and some applications. Itogi Nauki i Tekhniki, Probability Theory, Mathematical Statistics, Theoretical Cybernetics, vol. 24 (Russian), Akad. Nauk SSSR, Moscow (1986), pp. 53–110; Transl. J. Soviet Math. 42 (1988), 1766–1810.Google Scholar
[28]Smirnov, S. and Werner, W.Critical exponents for two-dimensional percolation. Math. Res. Lett. 8 (2001), 729–744.Google Scholar
[29]Tasaki, H.Hyperscaling inequalities for percolation. Commun. Math. Phys. 113 (1987), 49–65.Google Scholar
[30]Timár, Á.Boundary-connectivity via graph theory. Proc. Amer. Math. Soc. 141 (2013), 475–480.Google Scholar
[31]Werner, W.Lectures on two-dimensional critical percolation. Statistical Mechanics. IAS/Park City Math. Ser., vol. 16 (Amer. Math. Soc., Providence, RI, 2009), pp. 297–360.Google Scholar