TY - JOUR

T1 - The dimension-free structure of nonhomogeneous random matrices

AU - Latała, Rafał

AU - van Handel, Ramon

AU - Youssef, Pierre

N1 - Funding Information:
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.
Funding Information:
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. Gu?don 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.
Publisher Copyright:
© 2018, Springer-Verlag GmbH Germany, part of Springer Nature.

PY - 2018/12/1

Y1 - 2018/12/1

N2 - 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.

AB - 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.

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U2 - 10.1007/s00222-018-0817-x

DO - 10.1007/s00222-018-0817-x

M3 - Article

AN - SCOPUS:85053836946

VL - 214

SP - 1031

EP - 1080

JO - Inventiones Mathematicae

JF - Inventiones Mathematicae

SN - 0020-9910

IS - 3

ER -