The variance of the $\ell_p^n$-norm of the Gaussian vector, and Dvoretzky's theorem

Let $n$ be a large integer, and let $G$ be the standard Gaussian vector in $\mathbb R^n$. Paouris, Valettas and Zinn (2015) showed that for all $p\in[1,c\log n]$, the variance of the $\ell_p^n$-norm of $G$ is equivalent, up to a constant multiple, to $\frac{2^p}pn^{2/p-1}$, and for $p\in[C\log n,\infty]$, to $(\log n)^{-1}$. Here, $C,c>0$ are universal constants. That result left open the question of estimating the variance for $p$ logarithmic in $n$. In this paper, the question is resolved by providing a complete characterization of $\mathbf{Var}\|G\|_p$ for all $p$. It is shown that there exist two transition points (windows) in which the behavior of $\mathbf{Var}\|G\|_p$ changes significantly. Some implications of the results are discussed in the context of random Dvoretzky's theorem for $\ell_p^n$.