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→μ[tN](η))t∈(0, 1] in the thermodynamic limit N↑∞. Here μn(η) is the Gibbs measure in the finite volume {1,..., n} and η 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).