Free energy of a diluted spin glass model with quadratic Hamiltonian

TL;DR

This paper derives the limiting free energy of a diluted quadratic spin glass model using recursive distributional equations, providing the first explicit solution for this class of models at any temperature and external field.

Contribution

It introduces a novel method to compute the free energy of a diluted quadratic spin glass via recursive distributional equations, extending understanding of such models.

Findings

Limiting free energy expressed as an integral over random variables.

Identification of these variables as solutions to a recursive distributional equation.

First explicit free energy formula for the diluted Shcherbina-Tirozzi model.

Abstract

We study a diluted mean-field spin glass model with a quadratic Hamiltonian. Our main result establishes the limiting free energy in terms of an integral of a family of random variables that are the weak limits of the quenched variances of the spins in the system with varying edge connectivity. The key ingredient in our argument is played by the identification of these random variables as the unique solution to a recursive distributional equation. Our results in particular provide the first example of the diluted Shcherbina-Tirozzi model, whose limiting free energy can be derived at any inverse temperature and external field.