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≠0 and βN=1+bN−1/3√(log N), our main Gaussian orthogonal ensemble (GOE) result is √(6/(log N))(FN,βN−N FE(βN)+(log N)/12)xrightarrowdG+√(3/2) b+TW1, where G is standard Gaussian, TW1 has the real Tracy--Widom law, and G is independent of TW1, and FE denotes the limit of spherical Sherrington--Kirkpatrick free-energy. Additionally, we show that in a moderately supercritical regime 2/(N1/3(βN−1))(FN,βN−N FE(βN)+(log N)/12)xrightarrowdTW1. 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.
Related
- Fluctuations of the Sherrington-Kirkpatrick free energy at critical temperature
- Free energy and fluctuations of generalized bipartite Sherrington-Kirkpatrick models and generalized Hopfield models
- Quantitative Parisi formulas and fluctuations in the Sherrington-Kirkpatrick model
- Critical Free-Energy Variance in the Sherrington-Kirkpatrick Model
- Sampling from the Sherrington-Kirkpatrick Model up to Criticality
BibTeX
@misc{cheng2026criticalwindow,
title = {{Critical-Window Fluctuations and Disorder Universality for the Sherrington--Kirkpatrick Model}},
author = {Yu Cheng and Song-Hao Liu and Qi-Man Shao and Jing-Yu Xu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.07446},
eprint = {2609.07446},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.07446},
}
RIS
TY - UNPB
TI - Critical-Window Fluctuations and Disorder Universality for the Sherrington--Kirkpatrick Model
AU - Yu Cheng
AU - Song-Hao Liu
AU - Qi-Man Shao
AU - Jing-Yu Xu
PY - 2026
DA - 2026/09/07
PB - arXiv
DO - 10.48550/arxiv.2609.07446
UR - https://arxiv.org/abs/2609.07446
AB - We establish free-energy fluctuation limits for the Ising Sherrington--Kirkpatrick model in the nonzero parts of its critical window. For fixed b≠0 and β_N=1+bN^−1/3√(log N), our main Gaussian orthogonal ensemble (GOE) result is √(6/(log N))(F_N,β_N−N FE(β_N)+(log N)/12)xrightarrowdG+√(3/2) b_+TW_1, where G is standard Gaussian, TW_1 has the real Tracy--Widom law, and G is independent of TW_1, and FE denotes the limit of spherical Sherrington--Kirkpatrick free-energy. Additionally, we show that in a moderately supercritical regime 2/(N^1/3(β_N−1))(F_N,β_N−N FE(β_N)+(log N)/12)xrightarrowdTW_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.
ER -