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Inventiones mathematicae

, Volume 214, Issue 3, pp 1031–1080 | Cite as

The dimension-free structure of nonhomogeneous random matrices

  • Rafał Latała
  • Ramon van Handel
  • Pierre Youssef
Article

Abstract

Let X be a symmetric random matrix with independent but non-identically distributed centered Gaussian entries. We show that
$$\begin{aligned} \mathbf {E}\Vert X\Vert _{S_p} \asymp \mathbf {E}\Bigg [ \Bigg (\sum _i\Bigg (\sum _j X_{ij}^2\Bigg )^{p/2}\Bigg )^{1/p} \Bigg ] \end{aligned}$$
for any \(2\le p\le \infty \), 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 \(\ell _2\). Along the way, we obtain optimal dimension-free bounds on the moments \((\mathbf {E}\Vert X\Vert _{S_p}^p)^{1/p}\) 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.

Mathematics Subject Classification

60B20 46B09 46L53 15B52 

Notes

Acknowledgements

R.L. was supported by the National Science Centre of Poland under Grant Number 2015/18/A/ST1/00553. R.v.H. was supported in part by NSF Grant CAREER-DMS-1148711 and by the ARO through PECASE award W911NF-14-1-0094. P.Y. was supported by Grant ANR-16-CE40-0024-01. P.Y. would like to thank O. Guédon for introducing him to the questions investigated in this paper. This work was conducted while the authors were in residence at the Mathematical Sciences Research Institute in Berkeley, California, supported by NSF Grant DMS-1440140. The hospitality of MSRI and of the organizers of the program on Geometric Functional Analysis is gratefully acknowledged.

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • Rafał Latała
    • 1
  • Ramon van Handel
    • 2
  • Pierre Youssef
    • 3
  1. 1.Institute of MathematicsUniversity of WarsawWarsawPoland
  2. 2.Fine Hall 207Princeton UniversityPrincetonUSA
  3. 3.Laboratoire de Probabilités, Statistique et ModélisationUniversité Paris-DiderotParis CEDEX 13France

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