The limit free energy of spin-glass models with convex interactions can be
represented as a variational problem involving an explicit functional. Models with
nonconvex interactions are much less well understood, and simple variational
formulas involving the same functional are known to be invalid in general.
We show here that a slightly weaker property of the limit free energy does
extend to nonconvex models. Indeed, under the assumption that the limit free
energy exists, we show that this limit can always be represented as a critical
value of the said functional. Up to a small perturbation of the parameters
defining the model, we also show that any subsequential limit of the law
of the overlap matrix is a critical point of this functional. We believe that
these results capture the fundamental conclusions of the nonrigorous replica
method.
