Metastates in disordered mean-field models: Random field and Hopfield models
Statistical mechanics of disordered systems deals with the construction of Gibbs probability measures that are themselves random variables on some probability space describing the `quenched disorder', i.e. typically the random interactions of the system. On of the main issues is to describe the thermodynamic limit, i.e. the limiting measures obtained by taking the system size to infinity. In the random case, what is the most appropriate probabilistic setting to formulate this problem? This important and long neglected issue has been addressed in a series of papers by Ch. Newman and D. Stein [see, e.g., \textit{Ch. M. Newman}, ``Topics in disordered systems'' (1997; Zbl 0897.60093)] who proposed two concepts of `metastates' (probability states on the space of Gibbs measures) as appropriate candidates to study the infinite volume limits of disordered systems. In the present paper the author gives explicit constructions of the two types of metastates proposed in the simplest examples: the random field version of the Curie-Weiss model, and the Hopfield model with finitely many patterns. It turns out that already in this small class of models, a variety of behaviours can be found, and a lot of insight in the general concept can be gained (the author even exhibits a counterexample to a conjecture of Newman and Stein concerning the relation of their two types of metastates). The reviewer would recommend reading of this paper to anyone interested in understanding the metastate concept.
scientific article; zbMATH DE number 1279580
scientific article