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 parallel to X parallel to(Sp) asymptotic to E[(Sigma i(Sigma j Xij2)p/2)1/p] for any 2 <= p <= infinity, where Sp denotes the p-Schatten class and the constants are universal. The right-hand side admits an explicit expression in terms of the variances of thematrix entries. This settles, in the case p = infinity, 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 parallel to X parallel to(Sp)) 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.