Phase transitions and statistical mechanics

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Phase transitions and statistical mechanics

Collaborator: S. R $\oe$ lly

Cooperation with: P. Dai Pra (Università degli Studi di Padova, Italy), P.-Y. Louis (Université Lille 1, France)

Description:

Probabilistic Cellular Automata (PCAs) are discrete time Markov chains with parallel updating and local updating rules ([1]). PCAs are natural stochastic algorithms for parallel computing and as such have become widely used numerical tools in a large number of fields.

A key question in the theory of PCAs concerns the classification of their invariant measures according to their nature as stationary, reversible, or Gibbsian measures. This question was addressed in [2] for general PCAs, and the results were illustrated in the context of a class of reversible PCAs that were introduced by Lebowitz, Maes, and Speer ([4]). In fact, it has been known for a long time ([3]) that if a PCA is reversible with respect to a Gibbs measure corresponding to a potential $\Phi$ , then all its reversible measures are Gibbsian with respect to the same potential. In [2], a similar statement is now proven for the set of stationary measures:

For a general PCA, if one shift invariant stationary measure is Gibbsian for a potential $\Phi$ , then all shift invariant stationary measures are Gibbsian w.r.t. the same potential $\Phi$ .This induces that for a class of local, shift invariant, non-degenerated, reversible PCAs the reversible measures coincide with the Gibbsian stationary ones.

Applying this general statements to a particular class of reversible PCAs, it is shown in [2], using contour arguments that, for sufficiently small values of the temperature parameter, phase transition occurs, that is there are several Gibbs measures w.r.t. $\Phi$ . Furthermore, unlike what happens with sequential updating, a Gibbs measure which is not stationary for the associated PCA is exhibited.

References:

D.A. DAWSON, Synchronous and asynchronous reversible Markov systems, Canad. Math. Bull., 17 (1974/75), pp. 633-649.
P. DAI PRA, P.-Y. LOUIS, S. R $\oe$ lly, Stationary measures and phase transition for a class of Probabilistic Cellular Automata, WIAS Preprint no. 700, 2001.
O. KOZLOV, N. VASILYEV, Reversible Markov chains with local interaction, in: Multicomponent Random Systems, R.L. Dobrushin, Y.G. Sinai, eds., Adv. Probab. Related Topics, 6, Dekker, New York, 1980, pp. 451-469.
J.L. LEBOWITZ, C. MAES, E.R. SPEER, Statistical mechanics of Probabilistic Cellular Automata., J. Statist. Phys., 59 (1990), pp. 117-170.