Fluctuations of the free energy of the spherical Sherrington-Kirkpatrick model

TL;DR

This paper studies the fluctuations of free energy in the spherical Sherrington-Kirkpatrick model, revealing Gaussian behavior at high temperatures and Tracy-Widom distribution at low temperatures, with a focus on universality.

Contribution

It demonstrates the different fluctuation laws in high and low temperature regimes and establishes universality of the limit law for the spherical SK model.

Findings

Gaussian fluctuations in high temperature regime

Tracy-Widom fluctuations in low temperature regime

Universality of the limit law across regimes

Abstract

The spherical Sherrington-Kirkpatrick model is a spherical mean field model for spin glass. We consider the fluctuations of the free energy at arbitrary non-critical temperature for the 2-spin model with no magnetic field. We show that in the high temperature regime the law of the fluctuations converges to the Gaussian distribution just like in the Sherrington-Kirtkpatrick model. We show, on the other hand, that the law of the fluctuations is given by the GOE Tracy-Widom distribution in the low temperature regime. The orders of the fluctuations are markedly different in these two regimes. A universality of the limit law is also proved.