High Temperature Asymptotics of Orthogonal Mean-Field Spin Glasses

TL;DR

This paper analyzes the high temperature behavior of free energy in orthogonal mean-field spin glasses, connecting it to spectral measures and confirming previous non-rigorous results for the ROM.

Contribution

It introduces a method relating free energy limits to spectral measures using spherical integrals, applicable to various disordered spin glass models.

Findings

Derived the high temperature free energy limit for orthogonal spin glasses.

Confirmed non-rigorous results for the Random Orthogonal Model (ROM).

Extended methods to SK and Gaussian Hopfield models.

Abstract

We evaluate the high temperature limit of the free energy of spin glasses on the hypercube with Hamiltonian $H_N(\sigma) = \sigma^T J \sigma$, where the coupling matrix $J$ is drawn from certain symmetric orthogonally invariant ensembles. Our derivation relates the annealed free energy of these models to a spherical integral, and expresses the limit of the free energy in terms of the limiting spectral measure of the coupling matrix $J$. As an application, we derive the limiting free energy of the Random Orthogonal Model (ROM) at high temperatures, which confirms non-rigorous calculations of Marinari et al. (1994). Our methods also apply to other well-known models of disordered systems, including the SK and Gaussian Hopfield models.