The SK model with a sparse variance profile: free energy and AMP algorithm for TAP equations at high temperature

TL;DR

This paper extends the SK model with a sparse variance profile, deriving free energy asymptotics at high temperature and applying an AMP algorithm to estimate the spin mean via TAP equations.

Contribution

It generalizes the classical SK model to include sparse, unstructured variance profiles and adapts AMP algorithms for high-temperature free energy and TAP equation solutions.

Findings

Derived asymptotic free energy at high temperature for the generalized SK model.

Developed an AMP algorithm to estimate the spin vector mean.

Adapted the dynamical approach to the new model setting.

Abstract

A generalization of the Sherrington-Kirkpatrick (SK) model for spin glasses is considered, in which the interaction matrix is endowed with a variance profile that has no particular structure an may be sparse. In the first part of this paper, an asymptotic equivalent of the free energy is derived at sufficiently high temperatures, regardless of the signature of the variance profile matrix. In the second part, the mean of the spin vector under the Gibbs measure is estimated using an Approximate Message Passing algorithm based on the Thouless-Anderson-Palmer equations. The dynamical approach of Adhikari et.al. (J. Stat. Phys., 2021), originally developed for the classical SK model, is adapted to the present setting to obtain these results.