On partition functions of Gaussian random variables

For β∈ R, we define the partition function Z_β(X) = ∑ᵢ₌₁N exp(βXᵢ) of a centered Gaussian random vector X=(X₁, …, XN) with E[Xᵢ²]= 1. For q ∈ R, we prove a complete phase transition for the generalized Lq-means of Z_β(X) at $q=1$. The centered Gaussian vector with covariance matrix ΔN 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 β≠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.