Journal of Statistical Physics

, Volume 78, Issue 5–6, pp 1325–1335

Two applications of percolation to cellular automata

  • Jeffrey E. Steif


The point of this paper is to show how ideas from percolation can be used to study the asymptotic behavior of some cellular automata systems. In particular, using these ideas, we prove that the Greenberg-Hastings and cyclic cellular automata models with three colors, threshold 2, and theL neighborhood are uniformly asymptotically locally periodic ind≥2 dimensions. We also show that every lattice point is eventually “controlled by a finite clock” in the standard Greenberg-Hastings and cyclic cellular automata models in two dimensions, which is a stronger description than the already known asymptotic behavior.

Key Words

Cellular automata percolation 


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  1. 1.
    R. Durrett, Multicolor particle systems with large threshold and range,J. Theoret. Prob. 4(5):127–152 (1992).Google Scholar
  2. 2.
    R. Durrett and D. Griffeath, Asymptotic behavior of excitable cellular automata,J. Exp. Math., to appear.Google Scholar
  3. 3.
    R. Durrett and J. Steif, Some rigorous results for the Greenberg-Hastings model,J. Theoret. Prob. 4(4):669–690 (1991).Google Scholar
  4. 4.
    R. Durrett and J. Steif, Fixation results for threshold voter models,Ann. Prob. 21(1):232–247 (1993).Google Scholar
  5. 5.
    R. Fisch, The one-dimensional cyclic cellular automaton: A system with deterministic dynamics that emulates an interacting particle system with stochastic dynamics,J. Theoret. Prob. 3(2):311–338 (1989).Google Scholar
  6. 6.
    R. Fisch, Clustering in the one-dimensional 3-color cyclic cellular automaton,Ann. Prob. 20(3):1528–1548 (1992).Google Scholar
  7. 7.
    R. Fisch and J. Gravner, The one dimensional Greenberg-Hastings model, in preparation.Google Scholar
  8. 8.
    R. Fisch, J. Gravner, and D. Griffeath, InCyclic Cellular Automata in Two Dimensions (Birkhauser, Basel).Google Scholar
  9. 9.
    R. Fisch, J. Gravner, and D. Griffeath, Metastability in the Greenberg-Hastings model,Ann. Appl. Prob. 3:935–967 (1993).Google Scholar
  10. 10.
    R. Fisch, J. Gravner, and D. Griffeath, Threshold-range scaling for excitable cellular automata,Stat. Computing 1:23–39 (1991).Google Scholar
  11. 11.
    R. Fisch and D. Griffeath, Excitel: A periodic wave modeling environment, Freewave (1991).Google Scholar
  12. 12.
    J. Gravner, Ring dynamics in the Green-Hastings model, preprint.Google Scholar
  13. 13.
    J. Gravner, Doctoral Dissertation, University of Wisconsin at Madison.Google Scholar
  14. 14.
    J. M. Greenberg and S. P. Hastings, Spatial patterns for discrete models of diffusion in excitable media,SIAM J. Appl. Math. 34(3):515–523 (1978).Google Scholar
  15. 15.
    J. Steif, The threshold voter automaton at a critical point,Ann. Prob. 22(3) 1994.Google Scholar
  16. 16.
    A. T. Winfree, E. M. Winfree, and H. Seifert, Organizing centers in a cellular excitable medium,Physica 17D:109–115 (1985).Google Scholar

Copyright information

© Plenum Publishing Corporation 1995

Authors and Affiliations

  • Jeffrey E. Steif
    • 1
  1. 1.Department of MathematicsChalmers University of TechnologyGothenbergSweden

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