The dimension-free structure of nonhomogeneous random matrices

Rafał Latała; Ramon van Handel; Pierre Youssef

doi:10.1007/s00222-018-0817-x

paper

The dimension-free structure of nonhomogeneous random matrices

arXiv:1711.00807 · doi:10.1007/s00222-018-0817-x

Abstract

Let $X$ be a symmetric random matrix with independent but non-identically distributed centered Gaussian entries. We show that $E ∥ X ∥_{S_{p}} ≍ E [(i \sum (j \sum X_{ij}^{2})^{p /2})^{1/ p}]$ for any $2 \leq p \leq \infty$ , 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 $ℓ_{2}$ . Along the way, we obtain optimal dimension-free bounds on the moments $(E ∥ X ∥_{S_{p}}^{p})^{1/ p}$ 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.

36 pages, 2 figures

The dimension-free structure of nonhomogeneous random matrices

Abstract

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