### 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 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= ∞, 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‖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 language | English (US) |
---|---|

Pages (from-to) | 1031-1080 |

Number of pages | 50 |

Journal | Inventiones Mathematicae |

Volume | 214 |

Issue number | 3 |

DOIs | |

State | Published - Dec 1 2018 |

### All Science Journal Classification (ASJC) codes

- Mathematics(all)

## Fingerprint Dive into the research topics of 'The dimension-free structure of nonhomogeneous random matrices'. Together they form a unique fingerprint.

## Cite this

*Inventiones Mathematicae*,

*214*(3), 1031-1080. https://doi.org/10.1007/s00222-018-0817-x