Gaussian Convexity Principles for Sharp Moderate Deviations of Gaussian Maxima and Critical SK Free Energy Variance

Abstract

In this paper, we establish a moderate deviation bound for Gaussian maxima and the variance asymptotics of the Sherrington-Kirkpatrick free energy at criticality based on Gaussian convexity. First, let $(X_1,\ldots,X_N)$ be centered Gaussian vector with $\operatorname{Var}(X_i)\leq 1$. Suppose that, for fixed $α\in(0,\sqrt 2)$ and $κ>0$, $\mathbb{E}\max_iX_i\geqα\sqrt{\log N}$ and $\mathbb{E}\max_iX_i+κ\sqrt{\log N}\leq\sqrt{2\log N}$. We prove that $$ \mathbb{P}\left(\max_iX_i\geq \mathbb{E}\max_iX_i+κ\sqrt{\log N}\right) \leq N^{-κ^2/(2-α^2)+o(1)}. $$ This answers a question of Ding, Eldan and Zhai. The exponent is sharp, as witnessed by an equicorrelated Gaussian field. Second, for the Sherrington--Kirkpatrick model at the critical inverse temperature $β_c=1/\sqrt2$, we prove $$ \operatorname{Var}\bigl(F_N(β_c)\bigr)=\frac16\log N+O(1). $$ Our argument provides the variance asymptotics at the critical temperature from an entropy perspective, via a route distinct from that of Du and Huang. 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--R'enyi 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.