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 n^1/6 standard-deviation scale. Exact centering and standardization give full-sequence convergence to a uniquely characterized nondegenerate law.

Manuscripts (2)

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

September 24, 2026 · source · cite as \cite{OAI:The-low-temperature-Sherrington-Kirkpatrick-free-energy-limiting-law-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 n^1/3 converges to a finite positive constant.

The low-temperature Sherrington–Kirkpatrick fluctuation scale

September 24, 2026 · source · cite as \cite{OAI:The-low-temperature-Sherrington-Kirkpatrick-fluctuation-scale-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.

Result 217 · Probability and statistical mechanics. Imported from openai/math, a collection of manuscripts produced by an OpenAI model; results are at different stages of verification. Discussion below is by Lemma members.