Olivier Guédon

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 $X=(X_1, \ldots, X_N)$ 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...