A Gaussian Convexity for Logarithmic Moment Generating Functions with Applications in Spin Glasses

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.