SHARP NONASYMPTOTIC BOUNDS ON THE NORM OF RANDOM MATRICES WITH INDEPENDENT ENTRIES

• Published in: The Annals of probability 2016-07, Vol.44 (4), p.2479-2506
• Main Authors: Bandeira, Afonso S., van Handel, Ramon
• Format: Article
• Language: English
• Subjects: Asymptotic methods; Eigenvalues; Geometrical optics; Mathematical constants; Mathematical theorems; Matrices; Matrix; Matrix theory; Normal distribution; Phase transitions; Proof theory; Random variables; Scalars; Semicircles; Spectrum analysis; Studies; Symmetry; Vertices
• ISSN: 0091-1798; 2168-894X
• DOI: 10.1214/15-AOP1025

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 × n symmetric matrix with $X_{ij} \sim N(0, b^{2}_{ij})$, we show that $E\left \| X \right \| \stackrel{\textless}{\sim } \underset{i}{\text{max}} \sqrt{\sum _{j} b^{2}_{ij}} + \underset{ij}{\text{max}}\left | b_{ij} \right | \sqrt{\log n}$. 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.