At critical temperature the SK free energy has variance (1/6) log N + O(1) and a Gaussian CLT, while the annealed overlap satisfies E⟨R_{1,2}²⟩ ≍ N^{-2/3}.
A Gaussian Convexity for Logarithmic Moment Generating Functions with Applications in Spin Glasses
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abstract
For any convex function $F$ of $n$-dimensional Gaussian vector $g$ with $\mathbb{E} e^{\lambda F(g)}<\infty$ for any $\lambda>0$, we show that $\lambda^{-1}\ln \mathbb{E} e^{\lambda F(g)}$ is convex in $\lambda\in\mathbb{R}$. Based on this convexity, we draw three major consequences. The first recovers a version of the Paouris-Valettas lower deviation inequality for Gaussian convex functions with an improved exponent. The second establishes a quantitative bound for the Dotsenko-Franz-M\'ezard conjecture arising from the study of the Sherrington-Kirkpatrick (SK) mean-field spin glass model, which states that in the absence of external field, the annealed free energy of negative replica is asymptotically equal to the free energy. The last further establishes the differentiability for this annealed free energy with respect to the negative replica variable at any temperature and external field.
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2026 2representative citing papers
The sharp MSE bound for the ℓ1-minimum-norm interpolator under isotropic Gaussian covariates is recovered via the geometry of symmetric Gaussian polytopes, without the convex Gaussian min-max theorem.
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Fluctuations of the Sherrington-Kirkpatrick free energy at critical temperature
At critical temperature the SK free energy has variance (1/6) log N + O(1) and a Gaussian CLT, while the annealed overlap satisfies E⟨R_{1,2}²⟩ ≍ N^{-2/3}.
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Minimum Norm Interpolation via The Local Theory of Banach Spaces: The Role of Gaussianity
The sharp MSE bound for the ℓ1-minimum-norm interpolator under isotropic Gaussian covariates is recovered via the geometry of symmetric Gaussian polytopes, without the convex Gaussian min-max theorem.