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Probability and statistical mechanics · Result 217

The low-temperature Sherrington–Kirkpatrick fluctuation law

2 papers

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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Papers 2

First page of The low-temperature Sherrington–Kirkpatrick free-energy limiting law

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

OpenAI · September 24, 2026

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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@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}
}
First page of The low-temperature Sherrington–Kirkpatrick fluctuation scale

The low-temperature Sherrington–Kirkpatrick fluctuation scale

OpenAI · September 24, 2026

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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@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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