Moderate Deviations for Gaussian Maxima and an Entropy Proof of Critical SK Free Energy Fluctuations

We study two problems for Gaussian maxima. First, let $(X_1,\ldots,X_N)$ be a centered Gaussian vector with $\operatorname{Var}(X_i)\leq 1$. If, for fixed $\alpha\in(0,\sqrt{2})$ and $\kappa>0$, one has $\mathbb{E}\max_i X_i\geq\alpha\sqrt{\log N}$ and $\mathbb{E}\max_i X_i+\kappa\sqrt{\log N}\leq\sqrt{2\log N}$, then \[ \mathbb{P}\left(\max_i X_i\geq \mathbb{E}\max_i X_i+\kappa\sqrt{\log N}\right) \leq N^{-\kappa^2/(2-\alpha^2)+o(1)}. \] This answers a question of Ding, Eldan and Zhai, and the exponent is attained by an equicorrelated Gaussian field. Second, for the Sherrington--Kirkpatrick model at the critical inverse temperature $\beta_c=1/\sqrt{2}$, we prove \[ \operatorname{Var}\bigl(F_N(\beta_c)\bigr)=\frac{1}{6}\log N+O(1). \]
Our proof gives an alternative proof of the variance asymptotics at the critical temperature from an entropy perspective, independently of the approach of Du and Huang \cite{DuHuang2026}. For the upper bound, we express the variance as an entropy under exponential tilting and identify this entropy with the Kullback--Leibler divergence of a Gaussian synchronization model. Its derivative is then bounded using the I-MMSE formula, information percolation, and estimates for the susceptibility of the critical Erdős--Rényi random graph. For the lower bound, we combine Gaussian convexity applied at the replica parameter with an estimate for inverse moments on the sphere and an identity relating GOE eigenvalue densities in consecutive dimensions.