On the Aizenman exponent in critical percolation

  • L. N. Shchur
  • T. Rostunov
Miscellaneous

Abstract

The probabilities of clusters spanning a hypercube of dimension two to seven along one axis of a percolation system under criticality were investigated numerically. We used a modified Hoshen-Kopelman algorithm combined with Grassberger’s “go with the winner” strategy for the site percolation. We carried out a finite-size analysis of the data and found that the probabilities confirm Aizenman’s proposal of the multiplicity exponent for dimensions three to five. A crossover to the mean-field behavior around the upper critical dimension is also discussed.

PACS numbers

64.60.Ak 64.60.Fr 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    M. Aizenman, Nucl. Phys. B 485, 551 (1997); cond-mat/ 9609240.CrossRefADSMATHMathSciNetGoogle Scholar
  2. 2.
    L. N. Shchur and S. S. Kosyakov, Int. J. Mod. Phys. C 8, 473 (1997); cond-mat/9702248.ADSGoogle Scholar
  3. 3.
    J. Cardy, J. Phys. A 31, L105 (1998); cond-mat/9705137.CrossRefADSMATHGoogle Scholar
  4. 4.
    B. Duplantier, Phys. Rev. Lett. 82, 880 (1999); cond-mat/9812439.ADSMathSciNetGoogle Scholar
  5. 5.
    L. N. Shchur, in Computer Simulation Studies in Condensed-Matter Physics XII, Ed. by D. P. Landau, S. P. Lewis, and H.-B. Schüttler (Springer-Verlag, Heidelberg, 2000); cond-mat/9906013.Google Scholar
  6. 6.
    P. Grassberger, Comput. Phys. Commun. 147, 64 (2002); cond-mat/0201313; P. Grassberger and W. Nadler, cond-mat/0010265.ADSMATHMathSciNetGoogle Scholar
  7. 7.
    D. Stauffer and A. Aharony, Introduction to Percolation Theory (Taylor & Francis, London, 1992).Google Scholar
  8. 8.
    A. Bunde and S. Havlin, in Fractals and Disordered Systems, Ed. by A. Bunde and S. Havlin (Springer-Verlag, Berlin, 1996, 2nd ed.).Google Scholar
  9. 9.
    P. Sen, Int. J. Mod. Phys. C 8, 229 (1997); cond-mat/9704112.ADSGoogle Scholar
  10. 10.
    J. Hoshen and R. Kopelman, Phys. Rev. B 14, 3438 (1976).CrossRefADSGoogle Scholar
  11. 11.
    L. N. Shchur, Comput. Phys. Commun. 121–122, 83 (1999); hep-lat/0201015.MathSciNetGoogle Scholar
  12. 12.
    S. Smirnov, C. R. Acad. Sci., Ser. I 333, 239 (2001); http: //www.math.kth.se/~stas/papers/percras.ps.MATHGoogle Scholar
  13. 13.
    M. E. J. Newman and R. M. Ziff, Phys. Rev. Lett. 85, 4104 (2000); cond-mat/0005264.CrossRefADSGoogle Scholar
  14. 14.
    C. D. Lorenz and R. M. Ziff, J. Phys. A 31, 8147 (1998); cond-mat/9806224.CrossRefADSGoogle Scholar
  15. 15.
    G. Paul, R. M. Ziff, and H. E. Stanley, Phys. Rev. E 64, 026115 (2001); cond-mat/0101136.Google Scholar
  16. 16.
    P. Grassberger, cond-mat/0202144.Google Scholar
  17. 17.
    R. M. Ziff, Phys. Rev. Lett. 69, 2670 (1992).CrossRefADSGoogle Scholar
  18. 19.
    G. Andronico, A. Coniglio, and S. Fortunato, hep-lat/0208009.Google Scholar

Copyright information

© MAIK "Nauka/Interperiodica" 2002

Authors and Affiliations

  • L. N. Shchur
    • 1
  • T. Rostunov
    • 1
  1. 1.Landau Institute for Theoretical PhysicsRussian Academy of SciencesChernogolovka, Moscow regionRussia

Personalised recommendations