Operator $\ell_p\to\ell_q$ norms of Gaussian matrices

Abstract: We confirm the conjecture posed by Gu\'edon, Hinrichs, Litvak, and Prochno in 2017 that $\mathbb{E}|(a_{ij}g_{ij}){i\le m, j\le n}\colon \ell_p^n \to \ell_q^m|$ is comparable, up to constants depending only on $p$ and $q$, to [ \max_i |(a{ij})j|{p^*} +\max_j |(a_{ij})i|{q} +\mathbb{E} \max_{i,j} |a_{ij}g_{ij}| ] provided that $1\le p \le 2\le q \le \infty$. This was known before only in the case $p=1$ or $q=\infty$, and in the spectral case $p=2=q$. We also reprove the conjecture in the case $p=2=q$ without using spectral theory (which was employed in the previously known proof).