Operator ${\mathrm{\ell }}_p\to {\mathrm{\ell }}_q$ norms of Gaussian matrices

We confirm the conjecture posed by Guédon, Hinrichs, Litvak, and Prochno in 2017 that $\mathbb{E}\mathrm{\parallel }(a_{ij}{g}_{ij}{)}_{i\le m,j\le n}\text{\hspace{0.17em}⁣}:{\mathrm{\ell }}_p^n\to {\mathrm{\ell }}_q^m\mathrm{\parallel }$ is comparable, up to constants depending only on $p$ and $q$, to $$\underset{i}{\max }\mathrm{\parallel }(a_{ij}{)}_j{\mathrm{\parallel }}_{p^{\ast }}+\underset{j}{\max }\mathrm{\parallel }(a_{ij}{)}_i{\mathrm{\parallel }}_q+\mathbb{E}\underset{i,j}{\max }\mathrm{\mid }{a}_{ij}{g}_{ij}\mathrm{\mid }$$ provided that $1\le p\le 2\le q\le \mathrm{\infty }$. This was known before only in the case $p=1$ or $q=\mathrm{\infty }$, 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).