An informal introduction to the Parisi formula

TL;DR

This informal note surveys the SK spin-glass model, the Parisi formula, and the surprising link to partial differential equations. It presents the Parisi variational problem and its interpretation via an ultrametric Gibbs measure, alongside rigorous results establishing the limit of the free energy in the SK setting. The text then discusses extensions to multipartite models, where Parisi-type formulas are not yet settled, and outlines a PDE framework using an enriched free energy to characterize the limit via Hamilton-Jacobi equations, including challenges posed by nonconvex interactions in bipartite systems. The PDE perspective provides a unifying lens to study limit Free energies, offering both conceptual insights and a road map for future rigorous developments.

Abstract

This note is an informal presentation of spin glasses and of the Parisi formula. We also discuss some models for which the Parisi formula is not well-understood, and some partial progress that relies upon a connection with partial differential equations.

Figures (4)

• Figure 1.1: A simple si-tuation with frustration. Here 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 2.1: Except for some choices of $(W_{ij})$ of small probability and for large $N$, samples from the Gibbs measure essentially behave as if they were sampled from an ultrametric space. An ultrametric space can be encoded on the leaves of a tree as in the picture above, where the blue points are at the same distance from one another; they are themselves all at the same distance from any green point; the blue, green and orange points are all at the same distance from any red point, etc.
• Figure 3.1: Elementary units are organized into two layers and only interact across layers.
• Figure 4.1: Each characteristic line offers us a "prediction" for what the limit free energy is. For small $t$, each point $(t,q)$ (e.g. the orange point) is reached by exactly one characteristic line, so we know the value of the free energy then. For large $t$, it could happen that multiple characteristics reach a point, as happens for the black point. We know that the limit free energy must be as prescribed by one of those characteristics, but we do not know which one.

Theorems & Definitions (2)

• Theorem 4.1: The Parisi formula as a Hamilton-Jacobi equation chen2022hamiltonmourrat2022parisimourrat2020extending
• Conjecture 4.2