Journal of Statistical Physics

, Volume 60, Issue 3–4, pp 323–332 | Cite as

Finite-size effects for some bootstrap percolation models

  • A. C. D. van Enter
  • Joan Adler
  • J. A. M. S. Duarte
Articles

Abstract

The consequences of Schonmann's new proof that the critical threshold is unity for certain bootstrap percolation models are explored. It is shown that this proof provides an upper bound for the finite-size scaling in these systems. Comparison with data for one case demonstrates that this scaling appears to give the correct asymptotics. We show that the threshold for a finite system of sizeL scales asO[ln(lnL)] for the isotropic model in three dimensions where sites that fail to have at least four neighbors are culled.

Key words

Bootstrap percolation critical exponents phase transition finite-size scaling simulation 

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References

  1. 1.
    R. H. Schonmann, On the behavior of some cellular automata related to bootstrap percolation, Preprint.Google Scholar
  2. 2.
    R. H. Schonmann, Critical points of two dimensional bootstrap percolation like cellular automata,J. Stat. Phys. 58:1239 (1990).Google Scholar
  3. 3.
    J. Chalupa, P. L. Leath, and G. R. Reich,J. Phys. C 12:L31 (1981); see also M. Pollak and I. Riess,Phys. Stat. Sol. b 69:K15 (1975).Google Scholar
  4. 4.
    D. Griffeath,Notices AMS 35:1472 (1988); G. Vichniac,Physica 10D:96 (1988); G. Vichniac, inDisordered Systems and Biological Organization, E. Bienenstock, F. Fogelman-Soulié, and G. Weisbuch, eds. (Springer, Heidelberg, 1986).Google Scholar
  5. 5.
    J. Adler and A. Aharony, inSTATPHYS 16, Abstracts (Boston, Massachusetts, 1986); J. Adler and A. Aharony,J. Phys. A 21:1387 (1988).Google Scholar
  6. 6.
    J. A. M. S. Duarte,Physica A 157:1075 (1989).Google Scholar
  7. 7.
    J. Straley, unpublished, cited in ref. 3.Google Scholar
  8. 8.
    H. Nakanishi and H. Takano,Phys. Lett. A 115:187 (1986).Google Scholar
  9. 9.
    R. Lenormand and C. Zarcone, inKinetics of Aggregation and Gelation, F. Family and D. P. Landau, eds. (Elsevier, Amsterdam, 1984), p. 177.Google Scholar
  10. 10.
    A. C. D. van Enter,J. Stat. Phys. 48:943 (1988).Google Scholar
  11. 11.
    M. Aizenman and J. L. Lebowitz,J. Phys. A 21:3801 (1988).Google Scholar
  12. 12.
    J. Adler, D. Stauffer, and A. Aharony,J. Phys. A 22:L297 (1989).Google Scholar
  13. 13.
    W. Ertel, K. Frobose, and J. Jackle,J. Chem. Phys. 88:5027 (1988); Frobose, K.J. Stat. Phys. 55:1285 (1989).Google Scholar
  14. 14.
    J. Adler, A. Aharony, Y. Meir, and A. B. Harris, Series study of percolation moments in general dimension,Phys. Rev. B, to appear.Google Scholar
  15. 15.
    S. S. Manna, D. Stauffer, and D. W. Heermann,Physica A 162:20 (1990).Google Scholar
  16. 16.
    G. Ahlers,Rev. Mod. Phys. 52:489 (1980).Google Scholar
  17. 17.
    J.-H. Chen, M. E. Fisher, and B. G. Nickel,Phys. Rev. Lett. 48:630; A. J. Liu and M. E. Fisher,Physica A 156:35 (1989).Google Scholar
  18. 18.
    J. Adler,J. Phys. A 16:3835 (1983).Google Scholar
  19. 19.
    J. Adler, M. Moshe, and V. Privman, inPercolation Structures and Processes, G. Deutscher, R. Zallen, and J. Adler, eds. (Adam Hilger, London, 1983).Google Scholar
  20. 20.
    H. Kesten,Commun. Math. Phys. 109:109 (1987).Google Scholar
  21. 21.
    R. Schonmann, private communication.Google Scholar
  22. 22.
    J. A. M. S. Duarte, in preparation.Google Scholar

Copyright information

© Plenum Publishing Corporation 1990

Authors and Affiliations

  • A. C. D. van Enter
    • 1
  • Joan Adler
    • 2
    • 3
  • J. A. M. S. Duarte
    • 4
    • 5
  1. 1.Institute of Theoretical PhysicsRUGGroningenThe Netherlands
  2. 2.Department of PhysicsTechnionHaifaIsrael
  3. 3.Department of Physics and AstronomyUniversity of Tel AvivRamat AvivIsrael
  4. 4.HLRZJülich
  5. 5.Institut für Theoretische PhysikKoln 41West Germany

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