The low-temperature Sherrington–Kirkpatrick fluctuation law

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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.

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What the papers state

Theorem 1.1 (Limiting law)

In The low-temperature Sherrington–Kirkpatrick free-energy limiting law, 24 September 2026

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For every fixed real \beta>1, there is a nondegenerate Borel probability measure \nu_\beta such that

X_n(\beta)\ \Longrightarrow\ \nu_\beta \qquad\text{as }n\to\infty\text{ through all integers}.

The measure \nu_\beta has mean zero, variance one, and a finite exponential moment in a neighborhood of the origin. It is the unique probability measure with the moments specified in Proposition 1.3 below.

Corollary 1.4 (SK–Curie–Weiss fluctuations below unit coupling)

In The low-temperature Sherrington–Kirkpatrick free-energy limiting law, 24 September 2026

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Fix \beta>1 and 0\le\gamma<1. Using the same independent standard Gaussian couplings as in (1.1), define the exponent and log partition function

\begin{gathered} \mathcal H_n^{\mathrm{CW}}(\sigma;\beta,\gamma) = \frac{\beta}{\sqrt n}\sum_{1\le i<j\le n}g_{ij}\sigma_i\sigma_j +\frac{\gamma}{2n}\left(\sum_{i=1}^n\sigma_i\right)^2,\\ F_n^{\mathrm{CW}}(\beta,\gamma) =\log\sum_{\sigma\in\{-1,1\}^n} e^{\mathcal H_n^{\mathrm{CW}}(\sigma;\beta,\gamma)}. \end{gathered}

Here \gamma is the ferromagnetic coupling already in the exponent. As n\to\infty through all integers,

\begin{gathered} \Var F_n^{\mathrm{CW}}(\beta,\gamma) \sim a_2(\beta)n^{1/3},\\ \frac{F_n^{\mathrm{CW}}(\beta,\gamma) -\mathbb E F_n^{\mathrm{CW}}(\beta,\gamma)} {\sqrt{\Var F_n^{\mathrm{CW}}(\beta,\gamma)}} \Longrightarrow\nu_\beta. \end{gathered}

Here a_2(\beta)>0 is the constant a_2 above at the same \beta. The centering and standard deviation are the perturbed model’s own exact finite-size quantities.

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Four more possible matches

Theorem 1.1 in The low-temperature Sherrington–Kirkpatrick fluctuation scale, 24 September 2026.
Proposition 1.3 (Moment specification) in The low-temperature Sherrington–Kirkpatrick free-energy limiting law, 24 September 2026.
Theorem 1.2 (Moment convergence at the fluctuation scale) in The low-temperature Sherrington–Kirkpatrick free-energy limiting law, 24 September 2026.
Corollary 1.5 (Upper disorder-chaos scales in zero-field SK) in The low-temperature Sherrington–Kirkpatrick free-energy limiting law, 24 September 2026.

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Papers

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

24 September 2026, current edition.

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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

24 September 2026, current edition.

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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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