Abstract
The problem of metastability for a stochastic dynamics with a parallel updating rule is addressed in the Freidlin–Wentzel regime, namely, finite volume, small magnetic field, and small temperature. The model is characterized by the existence of many fixed points and cyclic pairs of the zero temperature dynamics, in which the system can be trapped in its way to the stable phase. Our strategy is based on recent powerful approaches, not needing a complete description of the fixed points of the dynamics, but relying on few model dependent results. We compute the exit time, in the sense of logarithmic equivalence, and characterize the critical droplet that is necessarily visited by the system during its excursion from the metastable to the stable state. We need to supply two model dependent inputs: (1) the communication energy, that is the minimal energy barrier that the system must overcome to reach the stable state starting from the metastable one; (2) a recurrence property stating that for any configuration different from the metastable state there exists a path, starting from such a configuration and reaching a lower energy state, such that its maximal energy is lower than the communication energy.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
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)
Bovier, A., Eckhoff, M., Gayrard, V., Klein, M.: Metastability and low lying spectra in reversible Markov chains. Comm. Math. Phys. 228, 219–255 (2002)
Cassandro, M., Galves, A., Olivieri, E., Vares, M.E.: Metastable behavior of stochastic dynamics: A pathwise approach. J. Stat. Phys. 35, 603–634 (1984)
Cirillo, E.N.M.: A note on the metastability of the Ising model: the alternate updating case. J. Stat. Phys. 106, 335–390 (2002)
Cirillo, E.N.M., Nardi, F.R.: Metastability for the Ising model with a parallel dynamics. J. Stat. Phys. 110, 183–217 (2003)
Cirillo, E.N.M., Nardi, F.R., Polosa, A.D.: Magnetic order in the Ising model with parallel dynamics. Phys. Rev. E 64, 57103 (2001)
Dai Pra, P., Louis, P.Y., Roelly, S.: Stationary measures and phase transition for a class of probabilistic cellular automata, ESAIM Probab. Statist. 6, 89–104 (2002, electronic)
Derrida, B.: Dynamical phase transition in spin model and automata. In: van Beijeren, H. (ed.) Fundamental problem in Statistical Mechanics, vol. VII. Elsevier (1990)
Manzo, F., Nardi, F.R., Olivieri, E., Scoppola, E.: On the essential features of metastability: Tunnelling time and critical configurations. J. Stat. Phys. 115, 591–642 (2004)
Georges, A., Le Doussal, P.: From equilibrium spin models to probabilistic cellular automata. J. Stat. Phys. 54, 1011–1064 (1989)
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)
Olivieri, E., Vares, M.E.: Large Deviations and Metastability. Cambridge University Press, Cambridge (2004)
Rujan, P.: Cellular automata and statistical mechanical models J. Stat. Phys. 49, 139–222 (1987)
Stavskaja, O.N.: Gibbs invariant measures for Markov chains on finite lattices with local interactions. Math. USSR Sbornik 21, 395–411 (1973)
Toom, A.L., Vasilyev, N.B., Stavskaja, O.N., Mitjushin, L.G., Kurdomov, G.L., Pirogov, S.A.: Locally interacting systems and their application in biology. In: Lect. Notes in Math., vol. 653. Springer, Berlin (1978)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
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). https://doi.org/10.1007/s10955-008-9563-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10955-008-9563-6