Critical-Window Fluctuations and Disorder Universality for the Sherrington--Kirkpatrick Model

Abstract

We establish free-energy fluctuation limits for the Ising Sherrington--Kirkpatrick model in the nonzero parts of its critical window. For fixed $b\ne0$ and $\beta_N=1+bN^{-1/3}\sqrt{\log N}$, our main Gaussian orthogonal ensemble (GOE) result is \[ \sqrt{\frac6{\log N}}\left(F_{N,\beta_N}-N\,\mathrm{FE}(\beta_N)+\frac{\log N}{12}\right)\xrightarrow{d}G+\sqrt{\frac32}\,b_+TW_1, \] where $G$ is standard Gaussian, $TW_1$ has the real Tracy--Widom law, and $G$ is independent of $TW_1$, and $\mathrm{FE}$ denotes the limit of spherical Sherrington--Kirkpatrick free-energy. Additionally, we show that in a moderately supercritical regime \[ \frac{2}{N^{1/3}(\beta_N-1)}\left(F_{N,\beta_N}-N\,\mathrm{FE}(\beta_N)+\frac{\log N}{12}\right)\xrightarrow{d}TW_1. \] We also show that the above results remains valid for independent, not necessarily identically distributed, disorder matrices whose first three moments match the Gaussian law and whose fourth moments satisfy an averaged bound.