# Fluctuations of the overlap at low temperature in the 2-spin spherical SK model

## TL;DR

This paper analyzes the low-temperature fluctuations of the overlap in the 2-spin spherical SK model, revealing they are governed by eigenvalues of GOE matrices and follow the Airy1 distribution.

## Contribution

It provides an explicit characterization of the overlap fluctuations in the low-temperature phase using random matrix theory, connecting spin glass behavior to GOE eigenvalues.

## Key findings

- Overlap fluctuations are of order N^{-1/3}.
- Fluctuations are described by eigenvalues of GOE matrices.
- Limiting distribution is given by the Airy1 random point field.

## Abstract

We describe the fluctuations of the overlap between two replicas in the 2-spin spherical SK model about its limiting value in the low temperature phase. We show that the fluctuations are of order $N^{-1/3}$ and are given by a simple, explicit function of the eigenvalues of a matrix from the Gaussian Orthogonal Ensemble. We show that this quantity converges and describe its limiting distribution in terms of the Airy1random point field (i.e., the joint limit of the extremal eigenvalues of the GOE) from random matrix theory.