**Graduate Studies in Mathematics**

Volume: 172;
2016;
461 pp;
Hardcover

MSC: Primary 05; 15; 33; 35; 41; 47; 52; 60; 82;

Print ISBN: 978-0-8218-4841-8

Product Code: GSM/172

List Price: $89.00

AMS Member Price: $71.20

MAA member Price: $80.10

**Electronic ISBN: 978-1-4704-3208-9
Product Code: GSM/172.E**

List Price: $89.00

AMS Member Price: $71.20

MAA member Price: $80.10

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#### Supplemental Materials

# Combinatorics and Random Matrix Theory

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*Jinho Baik; Percy Deift; Toufic Suidan*

Over the last fifteen years a variety of
problems in combinatorics have been solved in terms of random matrix
theory. More precisely, the situation is as follows: the problems at
hand are probabilistic in nature and, in an appropriate scaling limit,
it turns out that certain key quantities associated with these
problems behave statistically like the eigenvalues of a (large) random
matrix. Said differently, random matrix theory provides a
“stochastic special function theory” for a broad and
growing class of problems in combinatorics. The goal of this book is
to analyze in detail two key examples of this phenomenon viz., Ulam's
problem for increasing subsequences of random permutations and domino
tilings of the Aztec diamond. Other examples are also described along
the way, but in less detail.

Techniques from many different areas in mathematics are needed to
analyze these problems. These areas include combinatorics, probability
theory, functional analysis, complex analysis, and the theory of
integrable systems. The book is self-contained, and along the way we
develop enough of the theory we need from each area that a general
reader with, say, two or three years experience in graduate school can
learn the subject directly from the text.

#### Readership

Graduate students and research mathematicians interested in applications of the theory of random matrices to problems in combinatorics.

#### Reviews & Endorsements

...[T]he book is carefully written and will serve as an excellent reference.

-- Terence Tao, Mathematical Reviews

The book covers exciting results and has a wealth of information.

-- Milós Bóna, MAA Reviews

The book is self-contained, and along the way, we develop enough of the theory we need from each area that a general reader with, say, two or three years experience in graduate school can learn the subject directly from the text.

-- Zentralblatt MATH

#### Table of Contents

# Table of Contents

## Combinatorics and Random Matrix Theory

- Cover Cover11
- Title page iii4
- Dedication v6
- Contents vii8
- Preface xi12
- Chapter 1. Introduction 114
- Chapter 2. Poissonization and de-Poissonization 1932
- Chapter 3. Permutations and Young tableaux 2740
- Chapter 4. Bounds of the expected value of ℓ_{𝑁} 7790
- Chapter 5. Orthogonal polynomials, Riemann-Hilbert problems, and Toeplitz matrices 95108
- Chapter 6. Random matrix theory 139152
- Chapter 7. Toeplitz determinant formula 165178
- Chapter 8. Fredholm determinant formula 187200
- Chapter 9. Asymptotic results 207220
- Chapter 10. Schur measure and directed last passage percolation 253266
- Chapter 11. Determinantal point processes 305318
- Chapter 12. Tiling of the Aztec diamond 317330
- Chapter 13. The Dyson process and Brownian Dyson process 377390
- Appendix A. Theory of trace class operators and Fredholm determinants 421434
- Appendix B. Steepest-descent method for the asymptotic evaluation of integrals in the complex plane 431444
- Appendix C. Basic results of stochastic calculus 437450
- Bibliography 445458
- Index 459472
- Back Cover Back Cover1478