Extending the Parisi formula along a Hamilton-Jacobi equation

TL;DR

This paper links the free energy of mixed p-spin spin glass models with an added Gaussian magnetic field to the solution of a Hamilton-Jacobi equation, providing a new representation for soft spin models.

Contribution

It establishes a novel connection between the free energy of enriched spin glass models and Hamilton-Jacobi equations, extending the Parisi formula framework.

Findings

Free energy converges to the Hopf-Lax solution of a Hamilton-Jacobi equation.

Provides a new representation of free energy for mixed p-spin models with soft spins.

Links spin glass theory with PDE methods for the first time in this context.

Abstract

We study the free energy of mixed $p$-spin spin glass models enriched with an additional magnetic field given by the canonical Gaussian field associated with a Ruelle probability cascade. We prove that this free energy converges to the Hopf-Lax solution of a certain Hamilton-Jacobi equation. Using this result, we give a new representation of the free energy of mixed $p$-spin models with soft spins.