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 derive two-sided bounds for the expected value of the p-Schatten norm of X for 2≤p≤∞, with constants not depending on p. 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 l_2. Along the way, we obtain optimal dimension-free bounds on the L_p norm of the p-Schatten norm of X 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.