Result 217, Probability and statistical mechanics

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 cβn1/3c_\beta n^{1/3}, with cβ>0c_\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.

New or sharp bound

The bigger picture

Why it matters

In the Sherrington-Kirkpatrick model, binary spins interact through random Gaussian couplings. The reported result describes how the log partition function, which summarizes configurations' statistical weights, varies between samples of those couplings at low temperature.

What changes?

The unreviewed manuscripts report the result for zero external magnetic field and every fixed inverse temperature beta greater than one. For n spins, the log partition function's variance is asymptotic to a finite positive constant depending on beta times n to the one-third power. Its standard deviation therefore grows as n to the one-sixth power. Subtracting its exact expectation and dividing by its exact standard deviation yields convergence to a uniquely characterized, nondegenerate distribution along all integer system sizes.

What does that help mathematicians do?

The claim goes beyond assigning an exponent to fluctuations: it supplies a limiting probability law after exact normalization. Convergence along all system sizes rules out different limiting distributions on different subsequences, while nondegeneracy means the normalized fluctuations do not collapse to a single value. Researchers gain a precise target for descriptions of low-temperature disorder, including both the magnitude of sample-to-sample variation and its limiting distribution.

Are there practical applications?

The immediate value is foundational for statistical mechanics: the result quantifies how random interactions affect a central equilibrium quantity in this spin-glass model. It could provide a reference for numerical studies of finite systems. The supplied statements do not establish convergence rates, so they do not say how large a simulation must be to approximate the limiting law accurately.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

2 manuscripts

The low-temperature Sherrington–Kirkpatrick free-energy limiting law

September 24, 2026 126 pages

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.

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Cite (BibTeX) 
@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}
}

The low-temperature Sherrington–Kirkpatrick fluctuation scale

September 24, 2026 80 pages

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 n1/6+o(1)n^{1/6+o(1)}. The same exponent describes its typical centered absolute fluctuations, establishing the predicted one-sixth exponent in this regime.

PDF LaTeX source
Cite (BibTeX) 
@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}
}

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An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.