## The Annals of Probability

### Sharp nonasymptotic bounds on the norm of random matrices with independent entries

#### Abstract

We obtain nonasymptotic bounds on the spectral norm of random matrices with independent entries that improve significantly on earlier results. If $X$ is the $n\times n$ symmetric matrix with $X_{ij}\sim N(0,b_{ij}^{2})$, we show that

$\mathbf{E}\Vert X\Vert \lesssim\max_{i}\sqrt{\sum_{j}b_{ij}^{2}}+\max_{ij}\vert b_{ij}\vert \sqrt{\log n}.$ This bound is optimal in the sense that a matching lower bound holds under mild assumptions, and the constants are sufficiently sharp that we can often capture the precise edge of the spectrum. Analogous results are obtained for rectangular matrices and for more general sub-Gaussian or heavy-tailed distributions of the entries, and we derive tail bounds in addition to bounds on the expected norm. The proofs are based on a combination of the moment method and geometric functional analysis techniques. As an application, we show that our bounds immediately yield the correct phase transition behavior of the spectral edge of random band matrices and of sparse Wigner matrices. We also recover a result of Seginer on the norm of Rademacher matrices.

#### Article information

Source
Ann. Probab., Volume 44, Number 4 (2016), 2479-2506.

Dates
Revised: March 2015
First available in Project Euclid: 2 August 2016

https://projecteuclid.org/euclid.aop/1470139147

Digital Object Identifier
doi:10.1214/15-AOP1025

Mathematical Reviews number (MathSciNet)
MR3531673

Zentralblatt MATH identifier
1372.60004

#### Citation

Bandeira, Afonso S.; van Handel, Ramon. Sharp nonasymptotic bounds on the norm of random matrices with independent entries. Ann. Probab. 44 (2016), no. 4, 2479--2506. doi:10.1214/15-AOP1025. https://projecteuclid.org/euclid.aop/1470139147

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