The dimension-free structure of nonhomogeneous random matrices

Abstract

Let X be a symmetric random matrix with independent but non-identically distributed centered Gaussian entries. We show that

$$\begin{aligned} \mathbf {E}\Vert X\Vert _{S_p} \asymp \mathbf {E}\Bigg [ \Bigg (\sum _i\Bigg (\sum _j X_{ij}^2\Bigg )^{p/2}\Bigg )^{1/p} \Bigg ] \end{aligned}$$

for any $2\le p\le \infty$, where $S_p$ denotes the p-Schatten class and the constants are universal. The right-hand side admits an explicit expression in terms of the variances of the matrix entries. This settles, in the case $p=\infty$, a conjecture of the first author, and provides a complete characterization of the class of infinite matrices with independent Gaussian entries that define bounded operators on $\ell _2$. Along the way, we obtain optimal dimension-free bounds on the moments $(\mathbf {E}\Vert X\Vert _{S_p}^p)^{1/p}$ that are of independent interest. We develop further extensions to non-symmetric matrices and to nonasymptotic moment and norm estimates for matrices with non-Gaussian entries that arise, for example, in the study of random graphs and in applied mathematics.