Browse › Probability and statistical mechanics › 217 The low-temperature Sherrington–Kirkpatrick fluctuation law
The low-temperature Sherrington–Kirkpatrick free-energy limiting law
Open PDFCite Report a problem Add a reference or comment Interested in working on this Start a discussion
Edit NOTES.md · Existing issues · Source on GitHub
Abstract
For every fixed inverse temperature β > 1, we prove that the zero-field Gaussian Ising Sherrington–Kirkpatrick free energy, centered by its expectation and divided by its standard deviation, converges in distribution to a nondegenerate law as the system size tends to infinity through all integers. We also prove that its variance divided by n1/3 converges to a finite positive constant.
If the PDF does not display here, open it directly: Open PDF.
Notes
No notes yet. Add a reference, a comment or your name with the buttons above, or edit NOTES.md on GitHub.
Automatically collected notes (unreviewed)
work in progress
Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.
Follow-up works
- P. M. Aronow and Patrick Lopatto, Lower and Upper Bounds for Glauber Dynamics from Typical States of the Low-Temperature Sherrington–Kirkpatrick Model (hexagon:2610.00402v1, 2026-10-20). Cites this paper's Theorems 1.1–1.2 (variance prefactor $a_2(\beta)>0$ and limiting law), notes these would sharpen its mean-bias lower bound to $c_\beta n^{-5/6}$, and states that it does not use this paper. [AI-generated per source]
- P. M. Aronow and Patrick Lopatto, An $N^{-7/10}$ Bound for the Finite-Size Bias of the Sherrington–Kirkpatrick Free Energy (hexagon:2610.00177v1, 2026-10-12). Section 1.6 derives conditional mean-bias bounds from both manuscripts of this family, including an $N^{-5/6}$ lower bound from the positive variance prefactor and an upper bound from this paper's uncentered comparison. [AI-generated per source] [cites earlier version] [family-level]
Formalization
- OpenAI, SKFluctuations/EventuallyNatPos.lean (OpenAI math release, 2026). The companion-scale theorem
main_of_positiveLaplaceWindowexplicitly assumesPositiveLaplaceWindow; its conclusion concerns fluctuation exponents, not this paper's variance prefactor or limiting law. [AI-generated per source] [family-level] - P. M. Aronow and Patrick Lopatto, fluctuations-palomar (Lean package accompanying hexagon:2610.00402v1, 2026-10-20). Formalizes the companion paper's exponent $\mathrm{sd}(F_n)=n^{1/6+o(1)}$, which this paper uses as an input, with the positive-Laplace-window input proved internally; it does not formalize the limiting law or the $n^{1/3}$ variance prefactor. [AI-generated per source] [family-level]
Related prior literature mentioned by others
- T. Aspelmeier, A. Billoire, E. Marinari, M. A. Moore, Finite size corrections in the Sherrington-Kirkpatrick model (J. Phys. A 41 (2008) 324008). Cited in hexagon:2610.00402v1, Section 1.4, as a physics derivation of variance of order $n^{1/3}$ for the total free energy, the scale whose prefactor this paper proves positive; not in this paper's bibliography.
- Elizabeth Collins-Woodfin and Han Gia Le, Free energy fluctuations in SK and related spin glass models: A literature survey (arXiv:2510.26960v3, 2025-11-07). Aronow and Lopatto recommend this survey in Section 1.4 of hexagon:2610.00402v1 for background on fluctuation regimes, spin spaces and external fields. [family-level]
- Partha S. Dey and Taegu Kang, Fluctuations for the Sherrington–Kirkpatrick spin glass model near the critical temperature (arXiv:2603.05636, 2026-03-05). Near-critical variance asymptotics and a Gaussian limit for $\beta_N\uparrow1$, a regime this paper's fixed-$\beta$ law does not cover; cited in hexagon:2610.00402v1, and different from the Dey–Kang ferromagnetic paper in this paper's bibliography.
- Yan Ru Pei, Upper deviations of the Sherrington-Kirkpatrick free energy: the fifth-order law and the onset of constrained Parisi measures (arXiv:2610.06480, 2026-10-05). Treats the fifth-order upper-deviation law behind the Parisi–Rizzo tail-matching heuristic, which this paper's introduction contrasts with its limiting law; cited in hexagon:2610.00402v1.
- Jinho Baik and Ji Oon Lee, Fluctuations of the free energy of the spherical Sherrington-Kirkpatrick model (arXiv:1505.07349v5, 2015-07-05). Section 1.4 of hexagon:2610.00402v1 contrasts its low-temperature spherical-model Tracy–Widom law and $n^{1/3}$ total-free-energy scale with the Ising family's $n^{1/6}$ scale. [family-level]
Possibly relevant literature (found by the agent, not checked)
- T. Aspelmeier, Free energy fluctuations and chaos in the Sherrington-Kirkpatrick model (Phys. Rev. Lett. 100 (2008) 117205). Links free-energy sample-to-sample fluctuations to bond chaos, which bears on this paper's corollary deriving upper disorder-chaos scales from the variance asymptotic; not in this paper's bibliography.
Elsewhere
- Hexagon: An $N^{-7/10}$ Bound for the Finite-Size Bias of the Sherrington-Kirkpatrick Free Energy (AI-generated; human understanding: some)
- Hexagon: Lower and Upper Bounds for Glauber Dynamics from Typical States of the Low-Temperature Sherrington–Kirkpatrick Model (AI-generated; human understanding: some)
- VibeMathed: Problem record
- Emergent Mind: family 217
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:The-low-temperature-Sherrington-Kirkpatrick-free-energy-limiting-law-September-24-2026,
author = {{OpenAI}},
title = {{The low-temperature Sherrington--Kirkpatrick free-energy limiting law}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/The-low-temperature-Sherrington-Kirkpatrick-free-energy-limiting-law-September-24-2026/The-low-temperature-Sherrington-Kirkpatrick-free-energy-limiting-law-September-24-2026.pdf}{OAI:The-low-temperature-Sherrington-Kirkpatrick-free-energy-limiting-law-September-24-2026}},
year = {2026}
}Family 217: The low-temperature Sherrington–Kirkpatrick fluctuation law
For every fixed inverse temperature β > 1, determines the fluctuation scale and limiting law of the zero-field Gaussian Sherrington–Kirkpatrick log partition function. Its variance is asymptotic to $c_\beta n^{1/3}$, with $c_\beta\gt 0$, confirming the predicted n1/6 standard-deviation scale. Exact centering and standardization give full-sequence convergence to a uniquely characterized nondegenerate law.
Other papers in this family:
- The low-temperature Sherrington–Kirkpatrick fluctuation scale · September 24, 2026