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Spin glasses and the Parisi formula

Jean-Christophe Mourrat

TL;DR

The paper addresses the problem of identifying the limit free energy for spin-glass models beyond the classical Sherrington–Kirkpatrick (SK) framework, including multi-type and bipartite variants. It surveys the Parisi formula for the SK model, then presents an alternative un-inverted variational representation $f(\beta)=\sup_{\alpha\in\mathbf{Mart}}\{ \beta\sqrt{2}\mathbb{E}[\alpha_1 B_1]-\mathbb{E}[\phi^*(\alpha_1)]-\beta^2\sup_{t\in[0,1]} \int_t^1 (s-\mathbb{E}[\alpha_s^2]) ds \}$, derived via convex duality, and connects this to finite-$N$ and algorithmic considerations. A Hamilton–Jacobi PDE perspective is then developed by enriching the free energy with ultrametric parameters (path $q\in\mathcal{Q}$), yielding a limit equation $\partial_t f-\int_0^1 (\partial_q f)^2=0$ (for convex $\xi$) whose viscosity solution admits a variational form, recovering Parisi as a special case and guiding extensions to non-convex models like the bipartite case. The author conjectures a general un-inverted variational representation for all models with covariance $\mathbb{E}[H_N(\sigma)H_N(\tau)]=N\xi(\frac{\sigma\cdot\tau}{N})$, and outlines a program—via concave reformulations and affine-envelopes—to overcome non-convexity and establish such representations, potentially resolving main open questions in non-classical spin-glass models. Overall, the work proposes a unifying variational/PDE framework that links Parisi-type formulas, un-inverted representations, and Hamilton–Jacobi equations to tackle a broad class of spin-glass problems with practical and algorithmic implications.

Abstract

Spin glasses are models of statistical mechanics in which a large number of simple elements interact with one another in a disordered fashion. One of the fundamental results of the theory is the Parisi formula, which identifies the limit of the free energy of a large class of such models. Yet many interesting models remain out of reach of the classical theory, and direct generalizations of the Parisi formula yield invalid predictions. I will report here on some partial progress towards the resolution of this problem, which also brings a new perspective on classical results.

Spin glasses and the Parisi formula

TL;DR

The paper addresses the problem of identifying the limit free energy for spin-glass models beyond the classical Sherrington–Kirkpatrick (SK) framework, including multi-type and bipartite variants. It surveys the Parisi formula for the SK model, then presents an alternative un-inverted variational representation , derived via convex duality, and connects this to finite- and algorithmic considerations. A Hamilton–Jacobi PDE perspective is then developed by enriching the free energy with ultrametric parameters (path ), yielding a limit equation (for convex ) whose viscosity solution admits a variational form, recovering Parisi as a special case and guiding extensions to non-convex models like the bipartite case. The author conjectures a general un-inverted variational representation for all models with covariance , and outlines a program—via concave reformulations and affine-envelopes—to overcome non-convexity and establish such representations, potentially resolving main open questions in non-classical spin-glass models. Overall, the work proposes a unifying variational/PDE framework that links Parisi-type formulas, un-inverted representations, and Hamilton–Jacobi equations to tackle a broad class of spin-glass problems with practical and algorithmic implications.

Abstract

Spin glasses are models of statistical mechanics in which a large number of simple elements interact with one another in a disordered fashion. One of the fundamental results of the theory is the Parisi formula, which identifies the limit of the free energy of a large class of such models. Yet many interesting models remain out of reach of the classical theory, and direct generalizations of the Parisi formula yield invalid predictions. I will report here on some partial progress towards the resolution of this problem, which also brings a new perspective on classical results.

Paper Structure

This paper contains 6 sections, 3 theorems, 51 equations, 2 figures.

Table of Contents

  1. The Parisi formula
  2. Un-inverting the Parisi formula
  3. Towards more general models
  4. A connection with Hamilton-Jacobi equations
  5. Conjectured un-inverted formula for general models
  6. Conclusion

Key Result

Theorem 1.1

For every $\beta \geqslant 0$, we have where $\mathcal{P}([0,1])$ denotes the space of probability measures on $[0,1]$, and $\Phi_\mu :[0,1]\times \mathbb{R} \to \mathbb{R}$ is the solution to

Figures (2)

  • Figure 1.1: A simple situation with frustration. The coefficients $(W_{ij})$ suggest to set $\sigma_i = \sigma_j$, $\sigma_i = \sigma_k$, and $\sigma_j = - \sigma_k$, but we cannot realize these three conditions simultaneously.
  • Figure 3.1: The graph of direct interactions in the bipartite model.

Theorems & Definitions (4)

  • Theorem 1.1: Parisi formula gue03pan.aompanTpaper
  • Theorem 2.1: Un-inverted Parisi formula mourrat2024uninverting
  • Theorem 4.1: Limit free energy via a Hamilton-Jacobi equation chen2022hamiltonmourrat2022parisimourrat2020extending
  • Conjecture 4.2