On the free energy of vector spin glasses with non-convex interactions

TL;DR

This paper investigates the limit free energy of vector spin glasses with non-convex interactions, showing it can be characterized as a critical point of a certain functional, extending understanding beyond convex cases.

Contribution

It establishes that the limit free energy for non-convex vector spin glasses can be represented as a critical point of a specific functional, generalizing known results from convex models.

Findings

Limit free energy is a critical value of the functional.

Subsequential limits of the overlap law are critical points.

Results align with predictions from the replica method.

Abstract

The limit free energy of spin-glass models with convex interactions can be represented as a variational problem involving an explicit functional. Models with non-convex 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 non-convex 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 non-rigorous replica method.