On partition functions of Gaussian random variables

Source abstract

For $β\in \mathbb{R}$, we define the partition function $$Z_β(X) = \sum_{i=1}^N \exp(βX_i)$$ 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 $Z_β(X)$ at $q=1$. The centered Gaussian vector with covariance matrix $Δ_N$ obtained from a regular simplex configuration of unit vectors maximizes these means for $q 1$. The two regimes are governed by different principles. For $q > 1$ {and $β\ne0$}, 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.