Spherical spin glass model with external field

TL;DR

This paper investigates the effects of an external field on the spherical 2-spin spin glass model, analyzing free energy, overlaps, and phase transitions using advanced mathematical techniques, with results confirmed by recent independent work.

Contribution

It provides a detailed analysis of the transition between models with and without external fields, including limiting behaviors and fluctuations of key quantities.

Findings

Computed limiting free energy and overlaps as external field vanishes

Identified different scaling regimes and their matchings

Discussed implications for susceptibility and Gibbs measure geometry

Abstract

We analyze the free energy and the overlaps in the 2-spin spherical Sherrington Kirkpatrick spin glass model with an external field for the purpose of understanding the transition between this model and the one without an external field. We compute the limiting values and fluctuations of the free energy as well as three types of overlaps in the setting where the strength of the external field goes to zero as the dimension of the spin variable grows. In particular, we consider overlaps with the external field, the ground state, and a replica. Our methods involve a contour integral representation of the partition function along with random matrix techniques. We also provide computations for the matching between different scaling regimes. Finally, we discuss the implications of our results for susceptibility and for the geometry of the Gibbs measure. Some of the findings of this paper are confirmed rigorously by Landon and Sosoe in their recent paper which came out independently and simultaneously.