The dimension-free structure of nonhomogeneous random matrices

Let X be a symmetric random matrix with independent but non-identically distributed centered Gaussian entries. We show that E‖X‖Sp≍E[(∑i(∑jXij2)p/2)1/p]for any 2 ≤ p≤ ∞, 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= ∞, 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 ℓ₂. Along the way, we obtain optimal dimension-free bounds on the moments (E‖X‖Spp)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.