The Annals of Probability

Finite Size Scaling in Three-Dimensional Bootstrap Percolation

Raphaël Cerf and Emilio N. M. Cirillo

Full-text: Open access

Abstract

We consider the problem of bootstrap percolation on a three-dimensional lattice and we study its finite size scaling behavior. Bootstrap percolation is an example of cellular automata defined on the $d$-dimensional lattice ${1,2,\ldots,L^d}$ in which each site can be empty or occupied by a single particle; in the starting configuration each site is occupied with probability $p$,occupied sites remain occupied forever, while empty sites are occupied by a particle if at least $\ell$ among their 2$d$ nearest neighbor sites are occupied. When $d$ is fixed, the most interesting case is the one $\ell=d:$ this is a sort of threshold, in the sense that the critical probability $p_c$ for the dynamics on the infinite lattice $\mathbb{Z}^d$ switches from zero to one when this limit is crossed. Finite size effects in the three-dimensional case are already known in the cases $\ell\leq2$; in this paper we discuss the case $\ell=3$ and we show that the finite size scaling function for this problem is of the form $f(L) =const/lnlnL$.We prove a conjecture proposed by A.C.D. van Enter.

Article information

Source
Ann. Probab., Volume 27, Number 4 (1999), 1837-1850.

Dates
First available in Project Euclid: 31 May 2002

Permanent link to this document
https://projecteuclid.org/euclid.aop/1022874817

Digital Object Identifier
doi:10.1214/aop/1022874817

Mathematical Reviews number (MathSciNet)
MR1742890

Zentralblatt MATH identifier
0960.60088

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 82B43: Percolation [See also 60K35]

Keywords
Cellular automata bootstrap percolation finite size scaling critical length

Citation

Cerf, Raphaël; Cirillo, Emilio N. M. Finite Size Scaling in Three-Dimensional Bootstrap Percolation. Ann. Probab. 27 (1999), no. 4, 1837--1850. doi:10.1214/aop/1022874817. https://projecteuclid.org/euclid.aop/1022874817


Export citation

References

  • [1] Adler,J. and Aharony,A. (1988). Diffusion percolation I. Infinite time limit and bootstrap percolation. J. Phys. A: Math. Gen. 21 1387-1404.
  • [2] Adler,J.,Stauffer,D. and Aharony,A. (1989). Comparison of bootstrap percolation models. J. Phys. A: Math. Gen. 22 L297-L301.
  • [3] Aizenman,M. and Lebowitz,J. L. (1988). Metastability effects in bootstrap percolation. J. Phys. A: Math. Gen. 21 3801-3813.
  • [4] Branco,N. S.,Dos Santos,R. R. and de Queiroz,S. L. A. (1984). Bootstrap percolation: a renormalization group approach. J. Phys. C 17 L373-L377; Khan, M. A., Gould, H. and
  • Chalupa, J. (1985). Monte Carlo renormalization group study of bootstrap percolation. J. Phys. C 18 L223-L228; Branco, N. S., de Queiroz, S. L. A. and Dos Santos, R. R.
  • (1986). Critical exponents for high density and bootstrap percolation. J. Phys. C 19 1909-1921.
  • [5] Chalupa,J.,Leath,P. L. and Reich,G. R. (1979). Bootstrap percolation on a Bethe lattice. J. Phys. C: Solid State Phys. 12 L31-L37.
  • [6] Griffeath,D. (1988). Cyclic random competition: a case history in experimental mathematics. Notices Amer. Math. Soc. 35 1472-1480.
  • [7] Kogut,P. M. and Leath,P. L. (1981). J. Phys. C: Solid State Phys. 14 3187-3194.
  • [8] Le Normand,R. and Zarcone,C. (1984). Kinetics of Aggregation and Gelation (F. Family and D. P. Landau, eds.) North-Holland, Amsterdam.
  • [9] Mountford,T. S. (1995). Critical length for semi-oriented bootstrap percolation. Stochastic Process. Appl. 56 185-205.
  • [10] Schonmann,R. H. (1990). Critical Points of two-dimensional bootstrap percolation-like cellular automata. J. Statist. Phys. 58 1239-1244.
  • [11] Schonmann,R. H. (1990). Finite size scaling behavior of a biased majority rule cellular automaton. Phys. A 167 619-627.
  • [12] Schonmann,R. H. (1992). On the behavior of some cellular automata related to bootstrap percolation. Ann. Probab. 20 174-193.
  • [13] Toffoli,T. and Margolus,N. (1987). Cellular Automata Machines. A New Environment for Modeling. MIT Press.
  • [14] Ulam,S. (1950). Random processes and transformations. Proc. Internat. Congr. Math. 264- 275.
  • [15] van Enter,A. C. D. (1987). Proof of Straley's argument for bootstrap percolation. J. Statist. Phys. 48 943-945.
  • [16] van Enter,A. C. D.,Adler,J. and Duarte,J. A. M. S. (1990). Finite-size effects for some bootstrap percolation models. J. Statist. Phys. 60 323-332.
  • [17] van Enter,A. C. D.,Adler,J. and Duarte,J. A. M. S. (1991). Addendum: Finite-size effects for some bootstrap percolation models. J. Statist. Phys. 62 505-506.
  • [18] Vichniac,G. Y. (1984). Simulating physics with cellular automata. Phys. D 10 96-116.
  • [19] von Neumann,J. (1966). Theory of Self-Reproducing Automata. Univ. Illinois Press, Urbana.
  • [20] Wolfram,S. (1983). Statistical mechanics of cellular automata. Rev. Modern Phys. 55, 601-
  • 644; Wolfram,S. (1986). Theory and Applications of Cellular Automata. World Scientific, Singapore.