Fluctuations of the Sherrington-Kirkpatrick free energy at critical temperature
Abstract
We consider the Sherrington-Kirkpatrick spin glass model at the critical inverse temperature β= 1 with zero external field. We prove that the free energy FN = FN,β=1 of this model has variance Var(FN) = 1/6 log N + O(1) , confirming a physics prediction of Aspelmeier \cite{aspelmeier2008free}, and that the centered and scaled FN satisfies a Gaussian CLT. We also identify the critical two-replica overlap scale, proving 𝔼 ⟨ R1,22⟩ asymp N−2/3 , as conjectured by Talagrand \cite{talagrand2011mean2}, together with a uniform exponential moment bound for N1/3 |R1,2|. The key input is a critical reweighted moment method, in the spirit of the ``small subgraph conditioning'' technique from probabilistic combinatorics, but capable of capturing diverging fluctuations. Through this reweighting, we relate the critical SK model to the BBP critical edge, which determines the overlap and fluctuation scales.
Related
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BibTeX
@misc{du2026fluctuations,
title = {{Fluctuations of the Sherrington-Kirkpatrick free energy at critical temperature}},
author = {Hang Du and Brice Huang},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2607.02172},
eprint = {2607.02172},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.02172},
}
RIS
TY - UNPB
TI - Fluctuations of the Sherrington-Kirkpatrick free energy at critical temperature
AU - Hang Du
AU - Brice Huang
PY - 2026
DA - 2026/11/02
PB - arXiv
DO - 10.48550/arxiv.2607.02172
UR - https://arxiv.org/abs/2607.02172
AB - We consider the Sherrington-Kirkpatrick spin glass model at the critical inverse temperature β= 1 with zero external field. We prove that the free energy F_N = F_N,β=1 of this model has variance Var(F_N) = 1/6 log N + O(1) , confirming a physics prediction of Aspelmeier \cite{aspelmeier2008free}, and that the centered and scaled F_N satisfies a Gaussian CLT. We also identify the critical two-replica overlap scale, proving 𝔼 ⟨ R_1,2^2⟩ asymp N^−2/3 , as conjectured by Talagrand \cite{talagrand2011mean2}, together with a uniform exponential moment bound for N^1/3 |R_1,2|. The key input is a critical reweighted moment method, in the spirit of the ``small subgraph conditioning'' technique from probabilistic combinatorics, but capable of capturing diverging fluctuations. Through this reweighting, we relate the critical SK model to the BBP critical edge, which determines the overlap and fluctuation scales.
ER -