TY - JOUR

T1 - The dimension-free structure of nonhomogeneous random matrices

AU - Latała, Rafał

AU - van Handel, Ramon

AU - Youssef, Pierre

N1 - 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

SN - 0020-9910

VL - 214

SP - 1031

EP - 1080

JO - Inventiones Mathematicae

JF - Inventiones Mathematicae

IS - 3

ER -