Metastates in disordered mean-field models II: The superstates

We continue to investigate the size dependence of disordered mean-field models with finite local spin space in more detail, illustrating the concept of "superstates" as recently proposed by Bovier and Gayrard. We discuss various notions of convergence for the behavior of the paths (t --> mu([tN])(eta))(t is an element of (0, 1]) in the thermodynamic limit N up arrow) infinity. Here mu(n)(eta) is the Gibbs measure in the finite volume {1,..., n} and eta is the disorder variable. In particular we prove refined convergence statements in our concrete examples, the Hopfield model with finitely many patterns (having continuous paths) and the Curie-Weiss random-field Ising model (having singular paths).