Parisi measures

In the Parisi theory of spin glasses, the limiting free energy of the system is computed by optimizing over a “functional order parameter”. In mathematical terms this amounts to construct certain functions F(μ) of a probability measure μ on [0,1] and to compute the infimum over μ. The study of the maps μ↦F(μ) is a challenging problem of functional analysis. Progress on this problem seems required for further advances in the theory of spin glasses. The main objective of this paper is to explain the functional analysis part of the problems to the reader with no background (or interest) in spin glasses. As a first step in the study of these functions F(μ), we prove certain differentiability properties, that allow in certain cases to interpret (as conjectured by physicists) the Parisi measure (i.e. the probability μ at which F(μ) is minimum) in terms of spin glasses.