On partition functions of Gaussian random variables
Abstract
For β ∈ R, we define the partition function Zβ(X) = ∑i=1N exp(β Xi) 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 Zβ(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.
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BibTeX
@misc{friedland2026partition,
title = {{On partition functions of Gaussian random variables}},
author = {Omer Friedland and Olivier Guédon and Fabien Souli},
howpublished = {HAL},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.38904},
eprint = {2609.38904},
archivePrefix = {arXiv},
url = {https://hal.science/hal-05764076v1},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - On partition functions of Gaussian random variables
AU - Omer Friedland
AU - Olivier Guédon
AU - Fabien Souli
PY - 2026
DA - 2026/09/25
PB - HAL
DO - 10.48550/arxiv.2609.38904
UR - https://hal.science/hal-05764076v1
AB - For β ∈ R, we define the partition function Z_β(X) = ∑_i=1^N exp(β X_i) of a centered Gaussian random vector X=(X_1, …, X_N) with E[X_i^2]= 1. For q ∈ 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, 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.
ER -