Overlap of a spherical spin glass model with microscopic external field

TL;DR

This paper rigorously proves the behavior of the overlap in a spherical spin glass model with a microscopic external field, using contour integral analysis and random matrix theory techniques.

Contribution

It provides a rigorous mathematical proof for the overlap behavior in the 2-spin spherical SK model with a microscopic external field, extending previous non-rigorous physics results.

Findings

Overlap behavior characterized at microscopic external field strength

Contour integral representation of the generating function derived

Asymptotic analysis using random matrix theory applied

Abstract

We examine the behavior of the 2-spin spherical Sherrington-Kirkpatrick model with an external field by analyzing the overlap of a spin with the external field. Previous research has noted that, at low temperature, this overlap exhibits dramatically different behavior in the presence of an external field as compared to the model with no external field. The transition between those two settings was examined in a recent physics paper by Baik, Collins-Woodfin, Le Doussal, and Wu as well as a recent math paper by Landon and Sosoe. Both papers focus on the setting in which the external field strength, $h$, approaches zero as the dimension, $N$, approaches infinity. In particular, the paper of Baik et al studies the overlap with a microscopic external field ($h\sim N^{-1/2}$) but without a rigorous proof. This paper aims to give a proof of that result. The proof involves representing the generating function of the overlap as a ratio of contour integrals and then analyzing the asymptotics of those contour integrals using results from random matrix theory.