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Published in:
Interpolation of Probability Measures on Graphs Read chapter Read first chapter

2017 | OriginalPaper | Chapter

Structured Random Matrices

Author : Ramon van Handel

Published in: Convexity and Concentration

Publisher: Springer New York

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Abstract

Random matrix theory is a well-developed area of probability theory that has numerous connections with other areas of mathematics and its applications. Much of the literature in this area is concerned with matrices that possess many exact or approximate symmetries, such as matrices with i.i.d. entries, for which precise analytic results and limit theorems are available. Much less well understood are matrices that are endowed with an arbitrary structure, such as sparse Wigner matrices or matrices whose entries possess a given variance pattern. The challenge in investigating such structured random matrices is to understand how the given structure of the matrix is reflected in its spectral properties. This chapter reviews a number of recent results, methods, and open problems in this direction, with a particular emphasis on sharp spectral norm inequalities for Gaussian random matrices.

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previous chapter Concentration of Measure Principle and Entropy-Inequalities
next chapter Rates of Convergence for Empirical Spectral Measures: A Soft Approach
Footnotes
1
The first inequality follows by noting that for every v ∈ B, choosing \(\tilde{v} \in B_{\varepsilon }\) such that \(\|v -\tilde{ v}\| \leq \varepsilon\), we have \(\vert \langle v,Xv\rangle \vert = \vert \langle \tilde{v},X\tilde{v}\rangle +\langle v -\tilde{ v},X(v +\tilde{ v})\rangle \vert \leq \vert \langle \tilde{v},X\tilde{v}\rangle \vert + 2\varepsilon \|X\|\).
 
2
For reasons that will become evident in the proof, it is essential to consider (complex) unitary matrices U 1 ,U 2 ,U 3 , despite that all the matrices A k and X are assumed to be real.
 
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Metadata
Title
Structured Random Matrices
Author
Ramon van Handel
Copyright Year
2017
Publisher
Springer New York
Book
Convexity and Concentration

Print ISBN: 978-1-4939-7004-9

Electronic ISBN: 978-1-4939-7005-6

Copyright Year: 2017

DOI
https://doi.org/10.1007/978-1-4939-7005-6_4

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