Operator ${\mathrm{\ell }}_p\to {\mathrm{\ell }}_q$ norms of random matrices with iid entries

We prove that for every $p,q\in [1,\mathrm{\infty }]$ and every random matrix $X=(X_{i,j}{)}_{i\le m,j\le n}$ with iid centered entries satisfying the regularity assumption $\mathrm{\parallel }{X}_{i,j}{\mathrm{\parallel }}_{2\rho }\le \alpha \mathrm{\parallel }{X}_{i,j}{\mathrm{\parallel }}_{\rho }$ for every $\rho \ge 1$, the expectation of the operator norm of $X$ from ${\mathrm{\ell }}_p^n$ to ${\mathrm{\ell }}_q^m$ is comparable, up to a constant depending only on $\alpha$, to $$m^{1\mathrm{/}q}\underset{t\in B_p^n}{\sup }\parallel \sum _{j=1}^n{t}_j{X}_{1,j}{\parallel }_{q\wedge \mathrm{Log}m}+n^{1\mathrm{/}{p}^{\ast }}\underset{s\in B_{q^{\ast }}^m}{\sup }\parallel \sum _{i=1}^m{s}_i{X}_{i,1}{\parallel }_{p^{\ast }\wedge \mathrm{Log}n}\mathrm{.}$$ We give more explicit formulas, expressed as exact functions of $p$, $q$, $m$, and $n$, for the asymptotic operator norms in the case when the entries $X_{i,j}$ are: Gaussian, Weibullian, log-concave tailed, and log-convex tailed. In the range $1\le q\le 2\le p$ we provide two-sided bounds under a weaker regularity assumption $(\mathbb{E}{X}_{1,1}^4{)}^{1\mathrm{/}4}\le \alpha (\mathbb{E}{X}_{1,1}^2{)}^{1\mathrm{/}2}$.