Existence and Uniqueness of Permutation-Invariant Optimizers for Parisi Formula

It has recently been shown in [arXiv:2310.06745] that, upon constraining the system to stay in a balanced state, the Parisi formula for the mean-field Potts model can be written as an optimization problem over permutation-invariant functional order parameters. In this paper, we focus on permutation-invariant mean-field spin glass models. After introducing a correction term in the definition of the free energy and without constraining the system, we show that the limit free energy can be written as an optimization problem over permutation-invariant functional order parameters. We also show that for some models this optimization problem admits a unique optimizer. In the case of Ising spins, the correction term can be easily removed, and those results transfer to the uncorrected limit free energy. We also derive an upper bound for the limit free energy of some nonconvex permutation-invariant models. This upper bound is expressed as a variational formula and is related to the solution of some Hamilton-Jacobi equation. We show that if no first order phase transition occurs, then this upper bound is equal to the lower bound derived in [arXiv:2010.09114]. We expect that this hypothesis holds at least in the high temperature regime. Our method relies on the fact that the free energy of any convex mean-field spin glass model can be interpreted as the strong solution of some Hamilton-Jacobi equation.