A Hopf-like formula for mean-field spin glass models

TL;DR

This paper extends the Parisi formula for mean-field spin glass models into a concave, Lipschitz functional on signed measures, deriving a dual 'Hopf-like' formula related to a Hamilton-Jacobi equation in Wasserstein space.

Contribution

It introduces a dual formulation of the Parisi functional using Fenchel-Moreau duality, leading to a Hopf-like formula for mean-field spin glass models.

Findings

Derived a dual 'Hopf-like' formula for the free energy limit.

Extended the Parisi functional to a concave, Lipschitz functional on signed measures.

Connected the formula to Hamilton-Jacobi equations in Wasserstein space.

Abstract

We study mean-field spin glass models with general vector spins and convex covariance function. For those models, it is known that the limit of the free energy can be written as the supremum of a functional, this is the celebrated Parisi formula. In this paper, we observe that the Parisi functional extends into a concave and Lipschitz functional on the set of signed measures. We use this fact and Fenchel-Moreau duality to derive an un-inverted version of the Parisi formula. Namely, we show that the limit of the free energy can be written as the infimum of a functional related to the Parisi functional. This un-inverted formula can be interpreted as a Hopf-like formula for some Hamilton-Jacobi equation in Wasserstein space.