Toward a Generalized Parisi Ansatz

Abstract

In the analysis of the Sherrington–Kirkpatrick and mixed p-spin models, a key role is played by the fact that the Hamiltonian of these models is a Gaussian process with the covariance given by a function of the overlap of spin configurations in {−1,+1} N .The distribution of such processes is invariant under orthogonal transformations and, as a result, the computation of the free energy can be reduced to the description of the asymptotic distributions of the overlaps, which, in some sense, encode the Gibbs measure up to orthogonal transformations. However, for other random Hamiltonians on {−1,+1} N , understanding the distribution of the overlaps is not sufficient and one would like to study the asymptotic distributions of all coordinates, or spins, of the configurations sampled from the Gibbs measure. In certain models, it is expected that the structure of these asymptotic distributions can be described by some particular realizations of the Ruelle probability cascades on a separable Hilbert space, but, in most cases, these predictions remain an open problem. In this chapter, we will describe an approach that, in some sense, proves these predictions in the setting of the mixed p-spin models. Unfortunately, again, the special Gaussian nature of the Hamiltonian will play a crucial role but at least, we will obtain new information beyond the distribution of the overlaps.