Journal of Statistical Physics

, Volume 146, Issue 4, pp 701–718 | Cite as

Sharp Asymptotics for Stochastic Dynamics with Parallel Updating Rule

  • F. R. Nardi
  • C. Spitoni


In this paper we study the metastability problem for a stochastic dynamics with a parallel updating rule; in particular we consider a finite volume Probabilistic Cellular Automaton (PCA) in a small external field at low temperature regime. We are interested in the nucleation of the system, i.e., the typical excursion from the metastable phase (the configuration with all minuses) to the stable phase (the configuration with all pluses), triggered by the formation of a critical droplet. The main result of the paper is the sharp estimate of the nucleation time: we show that the nucleation time divided by its average converges to an exponential random variable and that the rate of the exponential random variable is an exponential function of the inverse temperature β times a prefactor that does not scale with β. Our approach combines geometric and potential theoretic arguments.


Stochastic dynamics Probabilistic cellular automata Metastability Potential theory Dirichlet form Capacity 


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  1. 1.
    Bianchi, A., Bovier, A., Ioffe, D.: Sharp asymptotics for metastability in the random field Curie-Weiss model. Electron. J. Probab. 14, 1541–1603 (2009) MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Bigelis, S., Cirillo, E.N.M., Lebowitz, J.L., Speer, E.R.: Critical droplets in metastable probabilistic cellular automata. Phys. Rev. E 59, 3935 (1999) MathSciNetADSCrossRefGoogle Scholar
  3. 3.
    Bovier, A., Eckhoff, M., Gayrard, V., Klein, M.: Metastability and low lying spectra in reversible Markov chains. Commun. Math. Phys. 228, 219–255 (2002) MathSciNetADSMATHCrossRefGoogle Scholar
  4. 4.
    Bovier, A., den Hollander, F., Nardi, F.R.: Sharp asymptotics for Kawasaki dynamics on a finite box with open boundary. Probab. Theory Relat. Fields 135, 265–310 (2006) MATHCrossRefGoogle Scholar
  5. 5.
    Bovier, A., den Hollander, F., Spitoni, C.: Homogeneous nucleation for Glauber and Kawasaki dynamics in large volumes and low temperature. Ann. Probab. 38, 661–713 (2010) MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Berman, K.A., Konsowa, M.H.: Random paths and cuts, electrical networks, and reversible Markov chains. SIAM J. Discrete Math. 3, 311–319 (1990) MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Bovier, A.: Metastability: a potential theoretic approach. In: Proceedings of ICM, 2006. EMS Publishing House, Zürich, pp. 499–518 (2006) Google Scholar
  8. 8.
    Bovier, A., Nardi, F.R., Spitoni, C.: Sharp asymptotics for stochastic dynamics with parallel updating rule with self-interaction. EURANDOM Report 2011-08 Google Scholar
  9. 9.
    Cassandro, M., Galves, A., Olivieri, E., Vares, M.E.: Metastable behavior of stochastic dynamics: a pathwise approach. J. Stat. Phys. 35, 603–634 (1984) MathSciNetADSMATHCrossRefGoogle Scholar
  10. 10.
    Cirillo, E.N.M., Nardi, F.R.: Metastability for the Ising model with a parallel dynamics. J. Stat. Phys. 110, 183–217 (2003) MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Cirillo, E.N.M., Nardi, F.R., Spitoni, C.: Metastability for reversible probabilistic cellular automata with self-interaction. J. Stat. Phys. 132, 431–471 (2008) MathSciNetADSMATHCrossRefGoogle Scholar
  12. 12.
    Cirillo, E.N.M., Nardi, F.R., Spitoni, C.: Competitive nucleation in reversible probabilistic cellular automata. Phys. Rev. E 78, 040601 (2008) MathSciNetADSCrossRefGoogle Scholar
  13. 13.
    Derrida, B.: Dynamical phase transition in spin models and automata. In: van Beijeren, H. (ed.) Fundamental Problems in Statistical Mechanics, vol. VII. Amsterdam, Elsevier (1990) Google Scholar
  14. 14.
    den Hollander, F.: Three lectures on metastability under stochastic dynamics. In: Methods of Contemporary Mathematical Statistical Physics. Lecture Notes in Math., vol. 1970. Springer, Berlin (2009) CrossRefGoogle Scholar
  15. 15.
    Manzo, F., Nardi, F.R., Olivieri, E., Scoppola, E.: On the essential features of metastability: tunneling time and critical configurations. J. Stat. Phys. 115, 591–642 (2004) MathSciNetADSMATHCrossRefGoogle Scholar
  16. 16.
    Olivieri, E., Scoppola, E.: Markov chains with exponentially small transition probabilities: first exit problem from a general domain. I. The reversible case. J. Stat. Phys. 79, 613–647 (1995) MathSciNetADSMATHCrossRefGoogle Scholar
  17. 17.
    Olivieri, E., Scoppola, E.: Markov chains with exponentially small transition probabilities: first exit problem from a general domain. II. The general case. J. Stat. Phys. 84, 987–1041 (1996) MathSciNetADSMATHCrossRefGoogle Scholar
  18. 18.
    Olivieri, E., Vares, M.E.: Large Deviations and Metastability. Cambridge University Press, Cambridge (2004) MATHGoogle Scholar

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© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.EURANDOMEindhovenThe Netherlands
  2. 2.Department of Mathematics and Computer ScienceEindhoven University of TechnologyEindhovenThe Netherlands
  3. 3.Department of Medical Statistics and BioinformaticsLeiden University Medical CentreLeidenThe Netherlands

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