The low-temperature Sherrington–Kirkpatrick fluctuation law
OpenAI research collection · 2 manuscripts · Source checked 2026-10-19
OpenAI's result summary
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_β n^{1/3}, with c_β> 0, confirming the predicted n^(1/6) standard-deviation scale. Exact centering and standardization give full-sequence convergence to a uniquely characterized nondegenerate law.
Read the manuscripts
Short excerpts below come from OpenAI's abstracts. The full abstract and original PDF preserve the paper's precise scope.
MANUSCRIPT 01
The low-temperature Sherrington–Kirkpatrick free-energy limiting law
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…
Full source 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 n^(1/3) converges to a finite positive constant.
This exact PDF is not listed in the imported formalization catalogue. Family scope notes may contain further information.
The low-temperature Sherrington–Kirkpatrick fluctuation scale
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.
This exact PDF is not listed in the imported formalization catalogue. Family scope notes may contain further information.
New mathematical reasoning could help AI tackle harder scientific and computational problems. How reliably these skills transfer beyond mathematics will determine how much they contribute to general intelligence.