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Structured Random Matrices

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Book cover Convexity and Concentration

Part of the book series: The IMA Volumes in Mathematics and its Applications ((IMA,volume 161))

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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Notes

  1. 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. 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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Acknowledgements

The author warmly thanks IMA for its hospitality during the annual program “Discrete Structures: Analysis and Applications” in Spring 2015. The author also thanks Markus Reiß for the invitation to lecture on this material in the 2016 spring school in Lübeck, Germany, which further motivated the exposition in this chapter. This work was supported in part by NSF grant CAREER-DMS-1148711 and by ARO PECASE award W911NF-14-1-0094.

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Authors and Affiliations

  1. Princeton University, Sherrerd Hall 227, Princeton, NJ, 08544, USA

    Ramon van Handel

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  1. Ramon van Handel
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Correspondence to Ramon van Handel .

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Editors and Affiliations

  1. Department of Mathematics, Rutgers University, Piscataway, New Jersey, USA

    Eric Carlen

  2. Department of Mathematical Sciences, University of Delaware, Newark, Delaware, USA

    Mokshay Madiman

  3. Department of Mathematics, Case Western Reserve University, Cleveland, Ohio, USA

    Elisabeth M. Werner

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van Handel, R. (2017). Structured Random Matrices. In: Carlen, E., Madiman, M., Werner, E. (eds) Convexity and Concentration. The IMA Volumes in Mathematics and its Applications, vol 161. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-7005-6_4

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