The dimension-free structure of nonhomogeneous random matrices

Rafał Latała, Ramon van Handel, Pierre Youssef

Research output: Contribution to journalArticle

Abstract

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 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 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.

Original languageEnglish (US)
Pages (from-to)1031-1080
Number of pages50
JournalInventiones Mathematicae
Volume214
Issue number3
DOIs
StatePublished - Dec 1 2018

ASJC Scopus subject areas

  • Mathematics(all)

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