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

Hong-Bin Chen, Jean-Christophe Mourrat

TL;DR

This work extends the rigorous understanding of mean-field vector spin glasses to non-convex interactions by linking the limit free energy to a Hamilton–Jacobi functional. Assuming the limit exists, the authors prove that the limit value is realized as a critical value of an explicit Parisi-type functional, with the associated overlap law corresponding to a critical point. They develop a robust framework based on continuous Poisson–Dirichlet cascades, cavity methods, and Ghirlanda–Guerra identities to analyze the overlap structure and to connect the limit to a Hamilton–Jacobi equation, recovering the replica-method intuition in a rigorous setting. The results provide a principled variational and dynamic description of non-convex vector spin glasses, highlighting the role of characteristic lines in the solution structure and offering a pathway toward a full characterization of the limit free energy beyond convex settings.

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.

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

TL;DR

This work extends the rigorous understanding of mean-field vector spin glasses to non-convex interactions by linking the limit free energy to a Hamilton–Jacobi functional. Assuming the limit exists, the authors prove that the limit value is realized as a critical value of an explicit Parisi-type functional, with the associated overlap law corresponding to a critical point. They develop a robust framework based on continuous Poisson–Dirichlet cascades, cavity methods, and Ghirlanda–Guerra identities to analyze the overlap structure and to connect the limit to a Hamilton–Jacobi equation, recovering the replica-method intuition in a rigorous setting. The results provide a principled variational and dynamic description of non-convex vector spin glasses, highlighting the role of characteristic lines in the solution structure and offering a pathway toward a full characterization of the limit free energy beyond convex settings.

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.
Paper Structure (38 sections, 52 theorems, 405 equations)

This paper contains 38 sections, 52 theorems, 405 equations.

Table of Contents

  1. Introduction
  2. Informal summary of the main results
  3. Precise statement of the main results
  4. Related works
  5. Outline of the paper
  6. Explicit representation of the Gaussian field H_N
  7. Assumption on the support of P_1
  8. Differentiability of Lipschitz functions
  9. Definition and regularity properties of the free energy
  10. Continuous cascades and overlap approximations
  11. The free energy with continuous cascades
  12. Cavity calculations and ultrametricity
  13. Definitions and notation
  14. Hamiltonians and perturbation
  15. Notation for the overlaps
  16. ...and 23 more sections

Key Result

Theorem 1.1

If $\xi$ is a convex function over $S^{D}_+$, then for every $t\geqslant 0$ and $q \in \mathcal{Q}_2$, we have where $f$ is the solution to e.hj.

Theorems & Definitions (107)

  • Theorem 1.1
  • Theorem 1.2: Critical point representation
  • Theorem 1.3: Critical point identification
  • Theorem 1.4: Identification of the law of the overlap
  • Proposition 1.5: Uniqueness of critical point at high temperature
  • Proposition 1.6: Relevant critical points must be stable
  • Definition 2.1: Fréchet differentiability
  • Definition 2.2: Gateaux differentiability
  • Definition 2.3
  • Definition 2.4
  • ...and 97 more