A formula for the free energy of non-convex vector spin glasses

We consider vector mean-field spin glass models with independent and identically distributed spins without assuming the covariance function is convex. We identify the limit free energy as the minimum of a functional that can be written as the sum of a relative entropy and the solution of a Hamilton–Jacobi equation. The proof of this result relies on the observation that when the spin configurations of the system are constrained to have a given empirical measure, the corresponding free energy approximately satisfies a Hamilton–Jacobi equation with a uniformly bounded error.