On partition functions of Gaussian random variables

For $\beta \in \mathbb{R}$, we define the partition function $$Z_{\beta }(X)=\sum _{i=1}^N\exp (\beta X_i)$$ of a centered Gaussian random vector $X=(X_1,\dots ,X_N)$ with $\mathbb{E}[X_i^2]=1$. For $q\in \mathbb{R}$, we prove a complete phase transition for the generalized $L_q$-means of $Z_{\beta }(X)$ at $q=1$. The centered Gaussian vector with covariance matrix ${\mathrm{\Delta }}_N$ obtained from a regular simplex configuration of unit vectors maximizes these means for $q<1$, and minimizes them for $q>1$. The two regimes are governed by different principles. For $q>1$ {and $\beta \mathrm{\ne }0$}, the moment functional is globally strongly convex on the entire set of correlation matrices, with an explicit modulus of convexity and a quantitative centroid-shape stability estimate. For $q<1$, the strategy is different. We prove a universal comparison for log-concave profiles of reverse Brascamp–Lieb type. Specializing this result to the Gumbel profile yields a Laplace-transform comparison between the partition functions.