Skip to main content
SpringerLink
Log in

Exponential Decay of Correlations for Strongly Coupled Toom Probabilistic Cellular Automata

  • Published:
Journal of Statistical Physics Aims and scope Submit manuscript

Abstract

We investigate the low-noise regime of a large class of probabilistic cellular automata, including the North-East-Center model of Toom. They are defined as stochastic perturbations of cellular automata belonging to the category of monotonic binary tessellations and possessing a property of erosion. We prove, for a set of initial conditions, exponential convergence of the induced processes toward an extremal invariant measure with a highly predominant spin value. We also show that this invariant measure presents exponential decay of correlations in space and in time and is therefore strongly mixing.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Bennett, C.H., Grinstein, G.: Role of irreversibility in stabilizing complex and nonergodic behavior in locally interacting discrete systems. Phys. Rev. Lett. 55(7), 657–660 (1985)

    Article  ADS  Google Scholar 

  2. Berezner, S., Krutina, M., Malyshev, V.: Exponential convergence of Toom’s probabilistic cellular automata. J. Stat. Phys. 73(5–6), 927–944 (1993). doi:10.1007/BF01052816

    Article  MathSciNet  ADS  MATH  Google Scholar 

  3. Bramson, M., Gray, L.: A useful renormalization argument. In: Durrett, R., Kesten, H. (eds.) Random walks, Brownian Motion, and Interacting Particle Systems: A Festschrift in Honor of Frank Spitzer. Progress in Probability. Birkhäuser, Basel (1991)

    Google Scholar 

  4. de Maere, A.: Phase transition and correlation decay in coupled map lattices. Commun. Math. Phys. 297(1), 229–264 (2010). doi:10.1007/s00220-010-1041-8

    Article  ADS  MATH  Google Scholar 

  5. Diakonova, M., MacKay, R.: Mathematical examples of space-time phases. Int. J. Bifurc. Chaos 21(8), 2297–2304 (2011)

    Article  Google Scholar 

  6. 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), 149–164 (1971)

    Google Scholar 

  7. Keller, G., Liverani, C.: Uniqueness of the SRB measure for piecewise expanding weakly coupled map lattices in any dimension. Commun. Math. Phys. 262(1), 33–50 (2006)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  8. Lebowitz, J., Maes, C., Speer, E.: Statistical mechanics of probabilistic cellular automata. J. Stat. Phys. 59, 117–170 (1990)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  9. Makowiec, D.: Toom probabilistic cellular automata stationary states via simulations. Acta Phys. Pol. B 29(6), 1599–1607 (1998)

    ADS  Google Scholar 

  10. Makowiec, D.: Stationary states of Toom cellular automata in simulations. Phys. Rev. E 60(4), 3787–3796 (1999)

    Article  ADS  Google Scholar 

  11. Stavskaya, O., Piatetski-Shapiro, I.I.: On homogeneous nets of spontaneously active elements. Syst. Theory Res. 20, 75–88 (1971)

    Google Scholar 

  12. Toom, A.: Monotonic binary cellular automata. Probl. Inf. Transm. 12(1), 33–37 (1976)

    MathSciNet  Google Scholar 

  13. Toom, A.: Stable and attractive trajectories in multicomponent systems. Adv. Probab. Relat. Top. 6, 549–575 (1980)

    MathSciNet  Google Scholar 

  14. 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 

  15. Toom, A., Vasilyev, N.B., Stavskaya, O., Mityushin, L.G., Kurdyumov, G.L., Pirogov, S.A.: Discrete local Markov systems. In: Dobrushin, R., Kryukov, V., Toom, A. (eds.) Stochastic Cellular Systems: Ergodicity, Memory, Morphogenesis. Manchester University Press, Manchester (1990)

    Google Scholar 

  16. Vasilyev, N.B., Petrovskaya, M.B., Piatetski-Shapiro, I.I.: Simulation of voting with random errors. Autom. Remote Control 30(10), 1639–1642 (1969)

    Google Scholar 

Download references

Acknowledgements

The authors would like to thank Jean Bricmont, Carlangelo Liverani and Christian Maes for helpful comments and discussions. They also thank the referees for their constructive remarks and suggestions.

Augustin de Maere was partially supported by the Belgian IAP (Interuniversity Attraction Pole) program P6/02.

Lise Ponselet was supported by a grant from the Belgian F.R.S.-FNRS (Fonds de la Recherche Scientifique) as ‘Aspirant FNRS’.

Author information

Authors and Affiliations

  1. IRMP, Université catholique de Louvain, Chemin du cyclotron 2, bte L7.01.03, 1348, Louvain-la-Neuve, Belgium

    Augustin de Maere & Lise Ponselet

Authors
  1. Augustin de Maere
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Lise Ponselet
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Lise Ponselet.

Rights and permissions

Reprints and permissions

About this article

Cite this article

de Maere, A., Ponselet, L. Exponential Decay of Correlations for Strongly Coupled Toom Probabilistic Cellular Automata. J Stat Phys 147, 634–652 (2012). https://doi.org/10.1007/s10955-012-0487-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10955-012-0487-9

Keywords

Search

Navigation