**Proceedings of Symposia in Applied Mathematics**

Volume: 72;
2014;
176 pp;
Hardcover

MSC: Primary 11; 60; 15; 05; 62;

Print ISBN: 978-0-8218-9471-2

Product Code: PSAPM/72

List Price: $56.00

Individual Member Price: $44.80

**Electronic ISBN: 978-1-4704-1660-7
Product Code: PSAPM/72.E**

List Price: $56.00

Individual Member Price: $44.80

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# Modern Aspects of Random Matrix Theory

Share this page *Edited by *
*Van H. Vu*

The theory of random matrices is an amazingly
rich topic in mathematics. Random matrices play a fundamental role in
various areas such as statistics, mathematical physics, combinatorics,
theoretical computer science, number theory and numerical
analysis.

This volume is based on lectures delivered at the 2013 AMS Short
Course on Random Matrices, held January 6–7, 2013 in San Diego,
California.

Included are surveys by leading researchers in the field, written
in introductory style, aiming to provide the reader a quick and
intuitive overview of this fascinating and rapidly developing
topic. These surveys contain many major recent developments, such as
progress on universality conjectures, connections between random
matrices and free probability, numerical algebra, combinatorics and
high-dimensional geometry, together with several novel methods and a
variety of open questions.

#### Table of Contents

# Table of Contents

## Modern Aspects of Random Matrix Theory

- Cover Cover11 free
- Title page iii4 free
- Contents v6 free
- Preface vii8 free
- Lecture notes on the circular law 110 free
- Free probability and random matrices 3544
- Random matrix theory, numerical computation and applications 5362
- Recent developments in non-asymptotic theory of random matrices 8392
- 1. Introduction 8392
- 2. Notation and basic definitions 8594
- 3. Subgaussian random variables 8695
- 4. Invertibility of a rectangular random matrix 8897
- 5. Invertibility of a square matrix: absolutely continuous entries 91100
- 6. Arithmetic structure and the small ball probability 96105
- 7. Putting all ingredients together 101110
- 8. Short Khinchin inequality 104113
- 9. Random unitary and orthogonal perturbations 110119
- References 118127

- Random matrices: The universality phenomenon for Wigner ensembles 121130
- 1. Introduction 121130
- 2. Global and local semi-circular laws 123132
- 3. Fine-scale spectral statistics: the case of GUE 129138
- 4. Extending beyond the GUE case I. \index{heat flow} methods 133142
- 5. Extending beyond the GUE case II. Swapping and the \index{Four Moment Theorem} 136145
- 6. Sketch of proof of four moment theorem 143152
- 7. Applications 148157
- 8. Open questions 164173
- References 166175

- Index iv5
- Back Cover Back Cover1186

#### Readership

Graduate students and research mathematicians interested in the theory of random matrices and its relations to other areas in mathematics.