Convergence Time of Probabilistic Cellular Automata on the Torus

  • Lorenzo TaggiEmail author
Part of the Emergence, Complexity and Computation book series (ECC, volume 27)


Many probabilistic cellular automata (PCA) exhibit a transition from an ergodic to a non-ergodic regime. Namely, if the free parameter is above a certain critical threshold, the process converges to a state that does not depend on the initial state (ergodicity), whereas if the free parameter is below the threshold, then the process converges to a state that depends on the initial state (non-ergodicity). If one considers the corresponding model on a finite space, such a transition is not observed (the process is always ergodic), nevertheless the convergence time is “small” when the corresponding process on infinite space is ergodic and “large” when the corresponding process on infinite space is non-ergodic. We analyse this correspondence for Percolation PCA, a class of probabilistic cellular automata which are closely related to oriented percolation.


  1. 1.
    Bagnoli, F., Rechtman, R.: Topological bifurcations in a model society of reasonable contrarians. Phys. Rev. E 88, 062914 (2013)CrossRefGoogle Scholar
  2. 2.
    Balister, P., Bollobás, B., Kozma, R.: Large deviations for mean field models of probabilistic cellular automata. Random Struct. Algorithms 29(3), 399–415 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Balister, P., Bollobás, B., Johnson, J., Walters, M.: Random majority percolation. Random Struct. Algorithms 36(3), 315–340 (2010)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Bennet, C., Grinstein, G.: Role of irreversibility in stabilizing complex and nonergodic behavior in locally interacting discrete systems. Phys. Rev. Lett. 55(7), 657–666 (1985)CrossRefGoogle Scholar
  5. 5.
    Berezner, S., Krutina, M., Malyshev, V.: Exponential convergence of Toom’s probabilistic cellular automata. J. Stat. Phys. 73(5–6), 927–944 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bigelis, S., Cirillo, E.N.M., Lebowitz, J.L., Speer, E.R.: Critical droplets in metastable states of probabilistic cellular automata. Phys. Rev E. 59, 3935–3941 (1999)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Chassaing, P., Mairesse, J.: A non ergodic probabilistic cellular automaton with a unique invariant measure. Stoch. Process. Appl. 121(11), 2474–2487 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Cirillo, E.N.M., Nardi, F.R., Spitoni, C.: Metastability for reversible probabilistic cellular automata with self-interaction. J. Stat. Phys. 132, 431–447 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Dai Pra, P., Louis, P.-Y., Roelly, S.: Stationary measures and phase transition for a class of probabilistic cellular automata. ESAIM: Probab. Stat. 6, 89–104 (2002)Google Scholar
  10. 10.
    Dai Pra, P., Sartori, E., Tolotti, M.: Strategic interaction in trend-driven dynamics. J. Stat. Phys. 152(4), 724–741 (2013)Google Scholar
  11. 11.
    de Maere, A., Ponselet, L.: Exponential decay of correlations for strongly coupled Toom probabilistic cellular automata. J. Stat. Phys. 147(3), 634–652 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Depoorter, J., Maes, C.: Stavskaya’s measure is weakly Gibbsian. Markov Process. Relat. Fields 12(4), 791–804 (2006)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Diakonova, M., MacKay, R.: Mathematical examples of space-time phases. Int. J. Bifurc. Chaos 21(8), 791–804 (2006)Google Scholar
  14. 14.
    Dobrushin, R.: Markov processes with a large number of locally interacting components: existence of a limit process and its ergodicity. Probl. Inf. Transm. 7(2), 1490164 (1071)Google Scholar
  15. 15.
    Durrett, R.: Oriented percolation in two dimensions. Ann. Probab. 12(4), 929–1227 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Durrett, R.: Probability: Theory and Examples, 4th edn. Cambridge University Press, Cambridge (2010)CrossRefzbMATHGoogle Scholar
  17. 17.
    Durrett, R., Schonmann, R.H., Tanaka, N.I.: The contact process on a finite set. III: the critical case. Ann. Probab. 17(4), 1303–1321 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Fatès, N.: Asynchronism induces second-order phase transitions in elementary cellular automata. J. Cell. Autom. 4(1), 21–38 (2009)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Fatès, N.: A guided tour of asynchronous cellular automata. J. Cell. Autom. 9, 387–416 (2014)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Fatès, N., Morvan, M., Schabanel, N., Thierry, É.: Fully asynchronous behavior of double-quiescent elementary cellular automata. Theor. Comput. Sci. 362(1–3), 1–16 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Fernández, R., Toom, A.: Non-Gibbsianness of the invariant measure of non-reversible cellular automata with totally asymmetric noise. Asthérisque 287, 71–87 (2003)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Gray, L.: The critical behaviour of a class of simple interacting systems - a few answers and a lot of questions. In: Durret, R. (ed.) Particle Systems, Random Media and Large Deviations. Contemporary Mathematics, vol. 41, pp. 149–160. AMS, Providence (1985). Asthérisque 287, 71–87 (2003)Google Scholar
  23. 23.
    Hinrichsen, H.: Nonequilibrium Critical Phenomena and Phase Transitions into Absorbing States. Lectures Held at the International Summer School on Problems in Statistical Physics XI. Leuven, Belgium (2005)Google Scholar
  24. 24.
    Kozma, R., Puljic, M., Balister, P., Bollobas, B., Freeman, W.: Phase transitions in the neuropercolation model for neural population with mixed local and non-local interactions. Biol. Cybern. 92, 367–379 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Landman, K.A., Binder, B.J., Newgreen, D.F.: Modeling development and disease in our “second” brain. Cell. Autom. Lect. Notes Comput. Sci. 7495, 405–414 (2012)Google Scholar
  26. 26.
    Lebowitz, J., Maes, C., Speer, E.: Statistical mechanics of probabillistic cellular automata. J. Stat. Phys. 59, 117–170 (1990)CrossRefzbMATHGoogle Scholar
  27. 27.
    Liggett, T.M.: Interacting Particle Systems, 2nd edn. Springer, Berlin (2005)CrossRefzbMATHGoogle Scholar
  28. 28.
    Louis, P.Y.: Ergodicity of PCA: equivalence between spatial and temporal mixing conditions. Electron. Commun. Probab. 9, 119–131 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Mairesse, J., Marcovici, I.: Around probabilistic cellular automata. J. Theor. Comput. Sci. 559, 42–72 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Makowiec, D.: Modeling heart pacemaker tissue by a network of stochastic oscillatory cellular automata. In: Mauri, G., et al. (eds.) UCNC 2013. LNCS, vol. 7956, pp. 138–149 (2013)Google Scholar
  31. 31.
    Manzo, F., Nardi, F.R., Olivieri, E., Scoppola, E.: On the essential features of metastability: tunnelling time and critical configurations. J. Stat. Phys. 115(1–2), 591–642 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Mendoça, J.: Monte Carlo investigation of the critical behavior of Stavskaya’s probabilistic cellular automaton. Phys. Rev. E 83(1), 012102 (2011)CrossRefGoogle Scholar
  33. 33.
    Pearce, C.E.M., Fletcher, F.K.: Oriented site percolation phase transitions and probability bounds. J. Inequal. Pure Appl. Math. 6(5), 135 (2005)MathSciNetzbMATHGoogle Scholar
  34. 34.
    Ponselet, L.: Phase transitions in probabilistic cellular automata. Ph.D. thesis (2013). arXiv:1312.3612
  35. 35.
    Regnault, D.: Proof of a phase transition in probabilistic cellular automata. Developments in Language Theory, pp. 433–444 (2013)Google Scholar
  36. 36.
    Shnirman, M.: On the problem of ergodicity of a Markov chain with infinite sets of states. Probl. Kibern. 20, 115–124 (1968)Google Scholar
  37. 37.
    Stavskaja, O.N.: Gibbs invariant measures for Markov chains on finite lattices with local interaction. Mat. Sbornik 21, 395 (1976)CrossRefzbMATHGoogle Scholar
  38. 38.
    Stavskaya, O., Piatetski-Shapiro, I.: On homogeneous nets of spontaneously active elements. Syst. Theory Res. 20, 75–88 (1971)Google Scholar
  39. 39.
    Taggi, L.: Critical probabilities and convergence time of percolation probabilistic cellular automata. J. Stat. Phys. 159(4), 853–892 (2015)Google Scholar
  40. 40.
    Toom, A.: A family of uniform nets of formal neurons. Sov. Math. Dokl. 9, 1338–1341 (1968)Google Scholar
  41. 41.
    Toom, A.: Stable and attractive trajectories in multicomponent systems. In: Dobrushin, R., Sinai, Y. (eds.) Multicomponent Random Systems. Advanced Probability Related Topics, vol. 6, pp. 549–575. Dekker, New York (1980)Google Scholar
  42. 42.
    Toom, A.: Cellular automata with errors: problems for students of probability. In: Snell, L. (ed.) Topics in Contemporary Probability and Its Applications. Probability and Stochastics Series. CRC Press, Boca Raton (1995)Google Scholar
  43. 43.
    Toom, A.: Contours, convex sets, and cellular automata. Notes for a Course Delivered at the 23th Colloquium of Brazilian Mathematics, Rio de Janeiro (2004)Google Scholar
  44. 44.
    Toom, A.: Ergodicity of cellular automata. Notes for a Course Delivered at Tartu University, Estonia (2013)Google Scholar
  45. 45.
    Toom, A., Vasilyev, N.B., Stavskaya, O.N., Mityushin, L.G., Kurdyumov, G.L., Pirogov, S.A.: Discrete local Markov systems. Stochastic Cellular Systems: Ergodicity, Memory, Morphogenesis. Manchester University Press, Manchester (1990)Google Scholar
  46. 46.
    Tomé, T., de Carvalho, K.C.: Stable oscillations of a predator-prey probabilistic cellular automaton: a mean-field approach. J. Phys. A.: Math. Theor. 40 (2007)Google Scholar
  47. 47.
    Varerstein, L., Leontovitch, A.: Invariant measures of certain Markov operators describing a homogeneous random medium. Probl. Inf. Transm. 6(1), 61–69 (1970)Google Scholar
  48. 48.
    Vasilyev, N., Petrovskaya, M., Piatetski-Shapiro, I.: Modelling of voting with random errors. Autom. Remote Control 10, 1632–1642 (Translated from Russian) (1970)Google Scholar

Copyright information

© Springer International Publishing AG 2018

Authors and Affiliations

  1. 1.Max Planck Institute for Mathematics in the SciencesLeipzigGermany
  2. 2.TU DarmstadtDarmstadtDeutschland

Personalised recommendations