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

Victor Issa

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.

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

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.

Paper Structure

This paper contains 20 sections, 34 theorems, 262 equations.

Table of Contents

  1. Introduction
  2. Preamble
  3. Main results
  4. Organization of the paper
  5. The enriched free energy
  6. Fenchel-Moreau duality in Wasserstein space
  7. Scalar models
  8. Variational representation for the initial condition
  9. Viscosity solution with linear initial condition
  10. Proof of the un-inverted Parisi formula
  11. Extension of concave functions
  12. Optimal transport of monotone probability measures
  13. Monotone probability measures on a compact set
  14. Building the extension
  15. Vector models
  16. ...and 5 more sections

Key Result

Theorem 1.1

If $\xi$ is convex on $S^D_+$, then we have for every $t > 0$

Theorems & Definitions (72)

  • Theorem 1.1: chenmourrat2023cavity
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 2.1: chenmourrat2023cavity
  • Remark 2.2
  • Definition 2.3
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • ...and 62 more