Central limit theorem near the critical temperature for the overlap in the 2-spin spherical SK model

TL;DR

This paper proves a central limit theorem for the overlap in the spherical SK model near the critical temperature, showing convergence in the high temperature phase with specific conditions on the inverse temperature.

Contribution

It establishes a CLT for the normalized overlap in the spherical SK model approaching the critical temperature at a polynomial rate.

Findings

CLT for normalized overlap in spherical SK model

Convergence holds almost surely with respect to disorder

Inverse temperature approaches critical value at rate > 1/3

Abstract

We prove a central limit theorem for the normalized overlap between two replicas in the spherical SK model in the high temperature phase. The convergence holds almost surely with respect to the disorder variables, and the inverse temperature can approach the criticial value at a polynomial rate with any exponent strictly greater than $1/3$.