On partition functions of Gaussian random variables

Abstract

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 $X=(X_1, \ldots, X_N)$ 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 $\Delta_N$ 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 $\beta\ne0$}, 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.