Critical-Window Fluctuations and Disorder Universality for the Sherrington--Kirkpatrick Model
Year: 2026
Type: preprint
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 $β_N=1+bN^{-1/3}\sqrt{\log N}$, our main Gaussian orthogonal ensemble (GOE) result is \[ \sqrt{\frac6{\log N}}\left(F_{N,β_N}-N\,\mathrm{FE}(β_N)+\frac{\log N}{12}\right)\xrightarrow{d}G+\sqrt{\frac32}\,b_+TW_1, \] 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 \[ \frac{2}{N^{1/3}(β_N-1)}\left(F_{N,β_N}-N\,\mathrm{FE}(β_N)+\frac{\log N}{12}\right)\xrightarrow{d}TW_1. \] 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.
Source: arXiv (Cornell University)
Authors: Cheng, Yu, Liu, Song-Hao, Shao, Qi-Man, Xu, Jing-Yu
Language: English
Subfield: Statistics and Probability
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7212075774