Legendre Duality of Spherical and Gaussian Spin Glasses

TL;DR

This paper explores the duality between Gaussian and spherical spin glass models through Legendre variational analysis, revealing their replica symmetry and discussing implications for the Gaussian Hopfield model.

Contribution

It introduces a Legendre duality framework linking the free energies of Gaussian and spherical spin glasses, demonstrating their replica symmetric nature.

Findings

Models are shown to be replica symmetric.

Legendre duality links free energies of the two models.

Application to Gaussian Hopfield model discussed.

Abstract

The classical result of concentration of the Gaussian measure on the sphere in the limit of large dimension induces a natural duality between Gaussian and spherical models of spin glass. We analyse the Legendre variational structure linking the free energies of these two systems, in the spirit of the equivalence of ensembles of statistical mechanics. Our analysis, combined with the previous work [4], shows that such models are replica symmetric. Lastly, we briefly discuss an application of our result to the study of the Gaussian Hopfield model.