On partition functions of Gaussian random variables

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 $q=1$. T