Annals of Probability

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

Afonso S. Bandeira and Ramon van Handel

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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

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

Received: August 2014
Revised: March 2015
First available in Project Euclid: 2 August 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60B20: Random matrices (probabilistic aspects; for algebraic aspects see 15B52)
Secondary: 46B09: Probabilistic methods in Banach space theory [See also 60Bxx] 60F10: Large deviations

Random matrices spectral norm nonasymptotic bounds tail inequalities


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.

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