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Stationary measures and phase transition for a class of Probabilistic Cellular Automata

Published online by Cambridge University Press:  15 November 2002

Paolo Dai Pra ,
Pierre-Yves Louis  and
Sylvie Rœlly
Paolo Dai Pra
Affiliation:
Dipartimento di Matematica Pura e Applicata, Università di Padova, Via Belzoni 7, 35131 Padova, Italy; [email protected].
Pierre-Yves Louis
Affiliation:
Laboratoire de Statistique et Probabilités, FRE 2222 du CNRS, UFR de Mathématiques, Université Lille 1, 59655 Villeneuve-d'Ascq Cedex, France; [email protected].
Sylvie Rœlly
Affiliation:
Weierstraß-Institut für Angewandte Analysis und Stochastik, Mohrenstraße 39, 10117 Berlin, Germany; [email protected]. : CMAP, UMR 7641 du CNRS, École Polytechnique, 91128 Palaiseau Cedex, France.
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Abstract

We discuss various properties of Probabilistic Cellular Automata, suchas the structure of the set of stationary measures and multiplicity ofstationary measures (or phase transition) for reversible models.

Keywords

Probabilistic Cellular Automatastationary measureGibbsmeasure.
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 6: New directions in Time Series Analysis (Guest Editor: Philippe Soulier)New directions in Time Series Analysis (Guest Editor: Philippe Soulier) , 2002 , pp. 89 - 104
Copyright
© EDP Sciences, SMAI, 2002

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References

Barron, A.R., The strong ergodic theorem for densities: Generalized Shannon-McMillan-Breiman theorem. Ann. Probab. 13 (1985) 1292-1303. CrossRef
Bigelis, S., Cirillo, E.N.M., Lebowitz, J.L. and Speer, E.R., Critical droplets in metastable states of probabilistic cellular automata. Phys. Rev. E 59 (1999) 3935-3941. CrossRef
P. Brémaud, Markov chains. Gibbs fields, Monte-Carlo simulation, and queues. Springer-Verlag, New York, Texts in Appl. Math. 31 (1999).
P. Dai Pra, Ph.D. Thesis. Rutgers University (1992).
D.A. Dawson, Synchronous and asynchronous reversible Markov systems. Canad. Math. Bull. 17 (1974/75) 633-649.
H.-O. Georgii, Gibbs measures and phase transitions. Walter de Gruyter & Co., Berlin, de Gruyter Stud. in Math. 9 (1988).
Goldstein, S., Kuik, R., Lebowitz, J.L. and Maes, C., From PCAs to equilibrium systems and back. Comm. Math. Phys. 125 (1989) 71-79. CrossRef
X. Guyon, Champs aléatoires sur un réseau. Modélisations, statistique et applications, Techniques stochastiques. Masson, Paris (1992).
Handa, K., Entropy production per site in (nonreversible) spin-flip processes. J. Statist. Phys. 83 (1996) 555-571. CrossRef
Holley, R., Free energy in a Markovian model of a lattice spin system. Comm. Math. Phys. 23 (1971) 87-99. CrossRef
O. Kozlov and N. Vasilyev, Reversible Markov chains with local interaction, Multicomponent random systems. Dekker, New York, Adv. Probab. Related Topics 6 (1980) 451-469.
Künsch, H., Nonreversible stationary measures for infinite interacting particle systems. Z. Wahrsch. Verw. Gebiete 66 (1984) 407-424. CrossRef
Künsch, H., Time reversal and stationary Gibbs measures. Stochastic Process. Appl. 17 (1984) 159-166. CrossRef
Lebowitz, J.L., Maes, C. and Speer, E.R., Statistical mechanics of probabilistic cellular automata. J. Statist. Phys. 59 (1990) 117-170. CrossRef
T.M. Liggett, Interacting particle systems, Vol. 276. Springer-Verlag, New York-Berlin (1985).
Lopez, F.J. and Sanz, G., Stochastic comparisons for general probabilistic cellular automata. Stat. Probab. Lett. 46 (2000) 401-410. CrossRef
Maes, C. and Shlosman, S.B., Ergodicity of probabilistic cellular automata: A constructive criterion. Comm. Math. Phys. 135 (1991) 233-251. CrossRef
C. Maes and S.B. Shlosman, When is an interacting particle system ergodic? Comm. Math. Phys. 151 (1993) 447-466.
Maes, C. and Vande Velde, K., The (non-) Gibbsian nature of states invariant under stochastic transformations. Physica A 206 (1994) 587-603. CrossRef
V.A. Malyshev and R.A. Minlos, Gibbs random fields, Cluster expansions. Kluwer Academic Publishers, Dordrecht, Math. Appl. 44 (1991).
F. Martinelli, Lectures on Glauber dynamics for discrete spin models, in Lectures on probability theory and statistics, Saint-Flour (1997) 93-191. Springer, Berlin, Lecture Notes in Math. 1717 (1999).
C. Preston, Random fields. Springer-Verlag, Berlin-New York, Lecture Notes in Math. 534 (1976).
A.L. Toom, N.B. Vasilyev, O.N. Stavskaya, L.G. Mityushin, G.L. Kurdyumov and S.A. Pirogov, Discrete local Markov systems, in Stochastic Cellular Systems: Ergodicity, memory, morphogenesis, edited by R.L. Dobrushin, V.I. Kryukov and A.L. Toom. Manchester University Press, Manchester (1990) 1-182.
N.B. Vasilyev, Bernoulli and Markov stationary measures in discrete local interactions, Locally interacting systems and their applications in biology. Pushchino (1976), edited by R.L. Dobrushin, V.I. Kryukov and A.L. Toom. Springer, Berlin, Lecture Notes in Math. 653 (1978).