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On the spectral norm of Gaussian random matrices


Author: Ramon van Handel
Journal: Trans. Amer. Math. Soc.
MSC (2010): Primary 60B20; Secondary 46B09, 60F10
DOI: https://doi.org/10.1090/tran/6922
Published electronically: May 30, 2017
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Abstract: Let $ X$ be a $ d\times d$ symmetric random matrix with independent but nonidentically distributed Gaussian entries. It has been conjectured by Latała that the spectral norm of $ X$ is always of the same order as the largest Euclidean norm of its rows. A positive resolution of this conjecture would provide a sharp understanding of the probabilistic mechanisms that control the spectral norm of inhomogeneous Gaussian random matrices. This paper establishes the conjecture up to a dimensional factor of order $ \sqrt {\log \log d}$. Moreover, dimension-free bounds are developed that are optimal to leading order and that establish the conjecture in special cases. The proofs of these results shed significant light on the geometry of the underlying Gaussian processes.


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Additional Information

Ramon van Handel
Affiliation: Fine Hall 208, Princeton University, Princeton, New Jersey 08544
Email: rvan@princeton.edu

DOI: https://doi.org/10.1090/tran/6922
Keywords: Random matrices, spectral norm, nonasymptotic bounds
Received by editor(s): August 24, 2015
Received by editor(s) in revised form: February 21, 2016
Published electronically: May 30, 2017
Additional Notes: The author was supported in part by NSF grant CAREER-DMS-1148711 and by the ARO through PECASE award W911NF-14-1-0094.
Dedicated: In memory of Evarist Giné
Article copyright: © Copyright 2017 American Mathematical Society