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eBook ISBN:  9781470432089 
Product Code:  GSM/172.E 
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Hardcover ISBN:  9780821848418 
eBook: ISBN:  9781470432089 
Product Code:  GSM/172.B 
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Hardcover ISBN:  9780821848418 
Product Code:  GSM/172 
List Price:  $135.00 
MAA Member Price:  $121.50 
AMS Member Price:  $108.00 
eBook ISBN:  9781470432089 
Product Code:  GSM/172.E 
List Price:  $85.00 
MAA Member Price:  $76.50 
AMS Member Price:  $68.00 
Hardcover ISBN:  9780821848418 
eBook ISBN:  9781470432089 
Product Code:  GSM/172.B 
List Price:  $220.00 $177.50 
MAA Member Price:  $198.00 $159.75 
AMS Member Price:  $176.00 $142.00 

Book DetailsGraduate Studies in MathematicsVolume: 172; 2016; 461 ppMSC: Primary 05; 15; 33; 35; 41; 47; 52; 60; 82
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 selfcontained, 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.
ReadershipGraduate students and research mathematicians interested in applications of the theory of random matrices to problems in combinatorics.

Table of Contents

Chapters

Chapter 1. Introduction

Chapter 2. Poissonization and dePoissonization

Chapter 3. Permutations and Young tableaux

Chapter 4. Bounds of the expected value of $\ell _N$

Chapter 5. Orthogonal polynomials, RiemannHilbert problems, and Toeplitz matrices

Chapter 6. Random matrix theory

Chapter 7. Toeplitz determinant formula

Chapter 8. Fredholm determinant formula

Chapter 9. Asymptotic results

Chapter 10. Schur measure and directed last passage percolation

Chapter 11. Determinantal point processes

Chapter 12. Tiling of the Aztec diamond

Chapter 13. The Dyson process and Brownian Dyson process

Appendix A. Theory of trace class operators and Fredholm determinants

Appendix B. Steepestdescent method for the asymptotic evaluation of integrals in the complex plane

Appendix C. Basic results of stochastic calculus


Additional Material

Reviews

...[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 selfcontained, 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


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

Chapters

Chapter 1. Introduction

Chapter 2. Poissonization and dePoissonization

Chapter 3. Permutations and Young tableaux

Chapter 4. Bounds of the expected value of $\ell _N$

Chapter 5. Orthogonal polynomials, RiemannHilbert problems, and Toeplitz matrices

Chapter 6. Random matrix theory

Chapter 7. Toeplitz determinant formula

Chapter 8. Fredholm determinant formula

Chapter 9. Asymptotic results

Chapter 10. Schur measure and directed last passage percolation

Chapter 11. Determinantal point processes

Chapter 12. Tiling of the Aztec diamond

Chapter 13. The Dyson process and Brownian Dyson process

Appendix A. Theory of trace class operators and Fredholm determinants

Appendix B. Steepestdescent method for the asymptotic evaluation of integrals in the complex plane

Appendix C. Basic results of stochastic calculus

...[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 selfcontained, 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