Structured Random Matrices

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.

46 pages; to appear in IMA Volume "Discrete Structures: Analysis and Applications" (Springer)

References in corpus (3)

• Sharp nonasymptotic bounds on the norm of random matrices with independent entries
• On the spectral norm of Gaussian random matrices
• Chaining, Interpolation, and Convexity

Cited by in corpus (7)

• Matrix Concentration Inequalities and Free Probability
• Structured networks and coarse-grained descriptions: a dynamical perspective
• Chaining, Interpolation, and Convexity II: The contraction principle
• Norms of Randomized Circulant Matrices
• Structured random matrices and cyclic cumulants: A free probability approach
• Moment bounds for large autocovariance matrices under dependence
• Norms of structured random matrices