Self-Organization of Random Cellular Automata: Four Snapshots

  • David Griffeath
Part of the NATO ASI Series book series (ASIC, volume 420)


We discuss four very simple random cellular automaton (CA) systems that self-organize over time. The first is a directed interface process which stabilizes in a coherent statistical equilibrium. The second is a model for excitable media: nucleating spiral cores lead to a locally periodic final state. The third model is a prototype for curvature-driven clustering. And the fourth illustrates the evolution of complex viable structures near phase boundaries in a parameterized family of non-linear population dynamics. For each CA we present a mix of rigorous results, conjectures, and empirical findings based on computer experimentation.

Key words

Cellular automaton interacting particle system interface excitable mediumn self-organization nucleation metastability artificial life 


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  1. [BCC]
    Bak, P., Chen, K., and Creutz, M. (1989). Self-organized criticality in the Game of Life. Nature 342, 780–782.ADSCrossRefGoogle Scholar
  2. [BB]
    Bennett, C. and Bourzutschky, M. (1991). Life not critical? Nature 350, 468.ADSCrossRefGoogle Scholar
  3. [BCG]
    Berlekamp, E., Conway, J., and Guy, R. (1982). Winning Ways for Your Mathematical Plays, Vol. 2. Academic Press, New York.zbMATHGoogle Scholar
  4. [BBC]
    Bidaux, R., Boccara, N., and Chaté, H. (1989). Order of the transition versus space dimension in a family of cellular automata. The Physical Review A 39, 3094–3105.ADSCrossRefGoogle Scholar
  5. [Dur1]
    Durrett, R. (1991). Probability: Theory and Examples. Wadsworth & Brooks/Cole, Pacific Grove, CA.zbMATHGoogle Scholar
  6. [Dur2]
    Durrett, R. (1993). Ten lectures on particle systems. To appear as 1993 Saint-Flour Probability Summer School Lecture Notes, Springer-Verlag, New York.Google Scholar
  7. [DG]
    Durrett, R. and Griffeath, D. Asymptotic behavior of excitable cellular automata. Journal of Experimental Mathematics 3, to appear.Google Scholar
  8. [DS]
    Durrett, R. and Steif, J. (1993). Fixation results for threshold voter systems. Annals of Probability 21, 232–247.MathSciNetzbMATHCrossRefGoogle Scholar
  9. [ES]
    Evans, L. C. and Spruck, J. (1993). Motion of level sets by mean curvature I. Journal of Differential Geometry, to appear.Google Scholar
  10. [FGGO]
    Fisch„ R., Gravner, J., and Griffeath, D. (1992). Cyclic cellular automata in two dimnensions. In Spatial Stochastic Processes. A festschrift in honor of the seventieth birthday of T. E. Harris (K. Alexander and J. Watkins, eds.), Birkhäuser, Boston, 171–185.Google Scholar
  11. [FGG1]
    Fisch, R., Gravner, J., and Griffeath, D. (1992). Threshold—range scaling of excitable cellular automata. Statistics and Computing 1, 23–39.CrossRefGoogle Scholar
  12. [FGG2]
    Fisch, R., Gravner, J., and Griffeath, D. (1993). Metastability in the Greenberg—Hastings Model. Annals of Applied Probability, to appear.Google Scholar
  13. [GK]
    Gandolfi, A. and Kesten, H. (1993). Greedy lattice animals II. Annals of Applied Probability, to appear.Google Scholar
  14. [GG1]
    Gravner, J. and Griffeath, D. Threshold growth dynamics. Transactions of the American Mathematical Society 341, to appear.Google Scholar
  15. [GG2]
    Gravner, J. and Griffeath, D. The Poisson—Voronoi limit for excitable cellular automata with rare nucleation. In preparation.Google Scholar
  16. [Gra]
    Gray, L. A strong law for the motion of interfaces in particle systems. In preparation.Google Scholar
  17. [GH]
    Greenberg, J. and Hastings, S. (1978). Spatial patterns for discrete models of diffusion in excitable media. SIAM Journal of Applied Mathematics 4, 515–523.MathSciNetCrossRefGoogle Scholar
  18. [Gri]
    Grimmett, G. (1989). Percolation. Springer-Verlag, New York.zbMATHGoogle Scholar
  19. [Lig]
    Liggett, T. M. (1985). Interacting Particle Systems. Springer-Verlag, New York.zbMATHCrossRefGoogle Scholar
  20. [Ruc]
    Rucker, R. (1990). CA-Lab (software). Autodesk, Sausalito, CA.Google Scholar
  21. [Tof]
    Toffoli, T. (1948). Integration of the phase-difference relations in asynchronous sequential networks. In Automata, Languages, and Programming (G. Ausiello and C. Böhm, ed.), Springer-Verlag, New York, 457–463.Google Scholar
  22. [TM]
    Toffoli, T. and Margolus, N. (1987). Cellular Automata Machines. MIT Press, Cambridge, Massachusetts.Google Scholar
  23. [WR]
    Weiner, N. and Rosenblueth, A. (1946). The mathematical foundation of the problem of conduction of impulses in a network of connected excitable elements, specifically in cardiac muscle. Archchive of the Institute of Cardiology, Mexico 16, 205–265.Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 1994

Authors and Affiliations

  • David Griffeath
    • 1
  1. 1.Department of MathematicsUniversity of WisconsinMadisonUSA

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