Advertisement

Journal of Statistical Physics

, Volume 128, Issue 6, pp 1383–1389 | Cite as

Finite-Size Effects for Anisotropic Bootstrap Percolation: Logarithmic Corrections

  • Aernout C. D. van Enter
  • Tim Hulshof
Open Access
Article

Abstract

In this note we analyse an anisotropic, two-dimensional bootstrap percolation model introduced by Gravner and Griffeath. We present upper and lower bounds on the finite-size effects. We discuss the similarities with the semi-oriented model introduced by Duarte.

Keywords

Bootstrap percolation Cellular automaton Finite size effects Metastability 

References

  1. 1.
    Adler, J.: Bootstrap percolation. Physica A 171, 452–470 (1991) CrossRefADSGoogle Scholar
  2. 2.
    Adler, J., Duarte, J.A.M.S., van Enter, A.C.D.: Finite-size effects for some bootstrap percolation models. J. Stat. Phys. 60, 323–332 (1990) CrossRefGoogle Scholar
  3. 3.
    Adler, J., Duarte, J.A.M.S., van Enter, A.C.D.: Finite-size effects for some bootstrap percolation models, addendum. J. Stat. Phys. 62, 505–506 (1991) CrossRefGoogle Scholar
  4. 4.
    Adler, J., Lev, U.: Bootstrap percolation: visualizations and applications. Braz. J. Phys. 33, 641–644 (2003) CrossRefGoogle Scholar
  5. 5.
    Aizenman, M., Lebowitz, J.L.: Metastability effects in bootstrap percolation. J. Phys. A: Math. Gen. 21, 3801–3813 (1988) CrossRefADSGoogle Scholar
  6. 6.
    Balogh, J., Bollobas, B.: Sharp thresholds in bootstrap percolation. Physica A 326, 305–312 (2003) MATHCrossRefADSGoogle Scholar
  7. 7.
    Biroli, G., Fisher, D.S., Toninelli, C.: Jamming percolation and glass transitions in lattice models. Phys. Rev. Lett. 96, 035702 (2006) CrossRefADSGoogle Scholar
  8. 8.
    Biroli, G., Fisher, D.S., Toninelli, C.: Cooperative behavior of kinetically constrained lattice gas models of glassy dynamics. J. Stat. Phys. 120, 167–238 (2005) CrossRefGoogle Scholar
  9. 9.
    Cerf, R., Cirillo, E.M.N.: Finite size scaling in three-dimensional bootstrap percolation. Ann. Probab. 27, 1833–1850 (1999) Google Scholar
  10. 10.
    Connelly, R., Rybnikov, K., Volkov, S.: Percolation of the loss of tension in an infinite triangular lattice. J. Stat. Phys. 105, 143–171 (2001) MATHCrossRefGoogle Scholar
  11. 11.
    De Gregorio, P., Lawlor, A., Bradley, P., Dawson, K.A.: Cellular automata with rare events; Resolution of an outstanding problem in the bootstrap percolation model. In: Cellular Automata. Amsterdam Proceedings, Lecture Notes in Computer Science, vol. 3305, pp. 365–374. Springer, Berlin (2004) Google Scholar
  12. 12.
    De Gregorio, P., Lawlor, A., Bradley, P., Dawson, K.A.: Clarification of the bootstrap percolation paradox. Phys. Rev. Lett. 93, 025501 (2004) CrossRefADSGoogle Scholar
  13. 13.
    Duarte, J.A.M.S.: Simulation of a cellular automaton with an oriented bootstrap rule. Physica A 157, 1075–1079 (1989) CrossRefADSGoogle Scholar
  14. 14.
    Gravner, J., Griffeath, D.: First passage times for threshold growth dynamics on ℤ2. Ann. Probab. 24, 1752–1778 (1996) (see also Griffeath’s webpage: http://psoup.math.wisc.edu/kitchen.html) MATHCrossRefGoogle Scholar
  15. 15.
    Gravner, J., Griffeath, D.: Scaling laws for a class of cellular automaton growth rules. In: Proceedings 1998 Erdös Center Workshop on Random Walks, pp. 167–186 (1999) Google Scholar
  16. 16.
    Gravner, J., Holroyd, A.: Slow convergence in bootstrap percolation. arXiv: 0705.1347 (2007) (see also Holroyd’s webpage: http://www.math.ubc.ca/holroyd/)
  17. 17.
    Holroyd, A.: Sharp metastability threshold for two-dimensional bootstrap percolation. Probab. Theory Relat. Fields 125, 195–224 (2003) MATHCrossRefGoogle Scholar
  18. 18.
    Holroyd, A.: The metastability threshold for modified bootstrap percolation in d dimensions. Electron. J. Probab. 11, 418–433 (2006) Google Scholar
  19. 19.
    Hulshof, W.J.T.: The similarities between an unbalanced and an oriented bootstrap percolation model. Groningen bachelor thesis (2007) Google Scholar
  20. 20.
    Kozma, R., Puljic, M., Balister, P., Bollobas, B., Freeman, W.J.: Phase transitions in the neuropercolation model of neural populations with mixed local and nonlocal interactions. Biol. Cybern. 92, 367–379 (2005) MATHCrossRefGoogle Scholar
  21. 21.
    Kirkpatrick, S., Wilcke, W.W., Garner, R.B., Huels, H.: Percolation in dense storage arrays. Physica A 314, 220–229 (2002) MATHCrossRefADSGoogle Scholar
  22. 22.
    Lee, I.H., Valentiniy, A.: Noisy contagion without mutation. Rev. Econ. Stud. 67, 47–56 (2000) MATHCrossRefGoogle Scholar
  23. 23.
    Lenormand, R.: Pattern growth and fluid displacement through porous media. Physica A 140, 114–123 (1986) CrossRefADSGoogle Scholar
  24. 24.
    Mountford, T.S.: Critical lengths for semi-oriented bootstrap percolation. Stoch. Proc. Appl. 95, 185–205 (1995) CrossRefGoogle Scholar
  25. 25.
    Mountford, T.S.: Comparison of semi-oriented bootstrap percolation models with modified bootstrap percolation. In: Boccara, N., Goles, E., Martinez, S. (eds.) Cellular Automata and Cooperative Systems. NATO ASI Proceedings, pp. 519–525. Kluwer Acadamic, Dordrecht (1993) Google Scholar
  26. 26.
    Ritort, F., Sollich, P.: Glassy dynamics of constrained models. Adv. Phys. 52, 219–342 (2003) CrossRefADSGoogle Scholar
  27. 27.
    Sabhapandit, S., Dhar, D., Shukla, P.: Hysteresis in the random-field Ising model and bootstrap percolation. Phys. Rev. Lett. 88, 197202 (2002) CrossRefADSGoogle Scholar
  28. 28.
    Schonmann, R.H.: Critical points of 2-dimensional bootstrap percolation-like cellular automata. J. Stat. Phys. 58, 1239–1244 (1990) CrossRefGoogle Scholar
  29. 29.
    Schonmann, R.H.: On the behavior of some cellular automata related to bootstrap percolation. Ann. Probab. 20, 174–193 (1992) MATHGoogle Scholar
  30. 30.
    Treaster, M., Conner, W., Gupta, I., Nahrstedt, K.: ContagAlert: using contagion theory for adaptive, distributed alert propagation. In: Fifth IEEE International Symposium on Network Computing and Applications, pp. 126–136 (2006) Google Scholar
  31. 31.
    van Enter, A.C.D.: Proof of Straley’s argument for bootstrap percolation. J. Stat. Phys. 48, 943–945 (1987) MATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  1. 1.Institute for Mathematics and Computing SciencesRijksuniversiteit GroningenGroningenThe Netherlands
  2. 2.GroningenThe Netherlands

Personalised recommendations