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The low-temperature Sherrington–Kirkpatrick fluctuation scale
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Abstract
For the zero-field Gaussian Sherrington–Kirkpatrick model at every fixed inverse temperature β > 1, we prove that the standard deviation of the log partition function is $n^{1/6+o(1)}$. The same exponent describes its typical centered absolute fluctuations, establishing the predicted one-sixth exponent in this regime.
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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). Restates this paper's main theorem as its Theorem 1.6, reproves the deduction from the positive-logarithmic-moment bound (its Section 20), and combines the variance exponent with a variance–bias inequality to get a mean-bias lower bound of order $n^{-5/6-\varepsilon}$ (its Corollary 1.7). [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 conditionally derives a mean-bias lower bound of order $N^{-5/6-\varepsilon}$ from this paper's variance exponent and compares consequences of the limiting-law companion. [AI-generated per source] [cites earlier version] [family-level]
Formalization
- OpenAI,
lean/OAI/Probability/SKFluctuationsin the release repository (openai/math, commit fd4aeeb). The release's Lean development for this result, whose theoremmain_of_positiveLaplaceWindowdeduces the standard-deviation, variance and typical-fluctuation exponents under the explicit hypothesisPositiveLaplaceWindow(the two-sided bound of this paper's Proposition 7.5, per hexagon:2610.00402v1, Section 1.5). [AI-generated per source] [family-level] - P. M. Aronow and Patrick Lopatto, fluctuations-palomar (Lean package accompanying hexagon:2610.00402v1, 2026-10-20). Formalizes this paper's standard-deviation, variance and typical-magnitude exponents with the Proposition 7.5 hypothesis proved internally; the README reports an axiom audit passed at commit 2fd7036, with a Comparator replay for one other result still pending. [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 argument that about $n^{1/6}$ RSB levels are resolved, giving variance of order $n^{1/3}$; 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 CLT for $\beta_N\uparrow1$, cited in hexagon:2610.00402v1 next to this paper's fixed-$\beta>1$ result; not 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). Derives the fifth-order constrained-Parisi cost behind the Parisi–Rizzo upper-deviation rate that this paper's history section discusses; cited in hexagon:2610.00402v1, Section 1.4.
- 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). A replica argument linking free-energy fluctuations to bond chaos that bounds the fluctuation exponent by 1/4, a physics bound that this paper's $n^{1/6+o(1)}$ theorem is consistent with; 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)
- Emergent Mind: family 217
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:The-low-temperature-Sherrington-Kirkpatrick-fluctuation-scale-September-24-2026,
author = {{OpenAI}},
title = {{The low-temperature Sherrington--Kirkpatrick fluctuation scale}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/The-low-temperature-Sherrington-Kirkpatrick-fluctuation-scale-September-24-2026/The-low-temperature-Sherrington-Kirkpatrick-fluctuation-scale-September-24-2026.pdf}{OAI:The-low-temperature-Sherrington-Kirkpatrick-fluctuation-scale-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.
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- The low-temperature Sherrington–Kirkpatrick free-energy limiting law · September 24, 2026