Metastates in disordered mean-field models: Random field and Hopfield models

We rigorously investigate the size dependence of disordered mean-field models with finite local spin space in terms of metastates. Thereby we provide an illustration of the framework of metastates for systems of randomly competing Gibbs measures. In particular we consider the thermodynamic limit of the empirical metastate 1/N Sigma(n=1)(N) delta(mu n(eta)), where mu(n)(eta) is the Gibbs measure in the finite volume {1, ..., n} and the frozen disorder variable eta is fixed. We treat explicitly the Hopfield model with finitely many patterns and the Curie-Weiss random field Ising model. In both examples in the phase transition regime the empirical metastate is dispersed for large N. Moreover, it does not converge for a.e. eta, but rather in distribution, for whose limits we give explicit expressions. We also discuss another notion of metastates, due to Aizenman and Wehr.