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Automates Cellulaires Probabilistes : mesures stationnaires, mesures de Gibbs associées et ergodicité

Abstract : Used in many scientific areas, the discrete time random dynamics called Probabilistic Cellular Automata (P.C.A.) are Markov stochastic processes with values in an infinite space S^G where S is a finite set and G an infinite graph. Here we assume G=Z^d. The main feature of these dynamics is the parallel, or synchronous, evolution of all the coordinates or interacting elementary components. We are first interested in the existence and uniqueness of stationary measures for non degenerated PCA dynamics, i.e. whose local behaviour is never deterministic. Based on results of Dai Pra, Kozlov, Künsch, Lebowitz, Vasilyev et al., we give for the class of reversible PCA dynamics precise relations between the sets of stationary measures, reversible measures and some Gibbs measures. For a typical parametrized family of reversible PCA dynamics, we prove the existence of a phase transition phenomenon and establish the behaviour of the different Gibbs measures under the dynamics' action. In particular, we exhibit non-stationary Gibbs measures. Secondly, we study the convergence to equilibrium for PCA dynamics which are attractive (so called ergodicity). One of our tools is a special coupling of these dynamics, preserving the stochastic order. When there is no phase transition, inspired by the ideas of Martinelli and Olivieri on Glauber dynamics, we prove the ergodicity of PCA dynamics, and more precisely, the convergence exponentially fast to the unique equilibrium state. This result truly improves previous ones existing in the litterature. Finally, we present some numerical simulations of reversible PCA dynamics, and, in particular, a parallel algorithm which converges to extremal Gibbs measures associated to the Ising model.
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https://tel.archives-ouvertes.fr/tel-00002203
Contributor : Pierre-Yves Louis <>
Submitted on : Wednesday, January 8, 2003 - 11:01:26 PM
Last modification on : Sunday, November 8, 2020 - 6:04:02 PM
Long-term archiving on: : Monday, September 20, 2010 - 12:07:52 PM

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  • HAL Id : tel-00002203, version 2

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Pierre-Yves Louis. Automates Cellulaires Probabilistes : mesures stationnaires, mesures de Gibbs associées et ergodicité. Mathématiques [math]. Université des Sciences et Technologie de Lille - Lille I, 2002. Français. ⟨tel-00002203v2⟩

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