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

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.