Hardcover ISBN:  9780821852859 
Product Code:  SURV/171 
632 pp 
List Price:  $117.00 
MAA Member Price:  $105.30 
AMS Member Price:  $93.60 
Electronic ISBN:  9781470413989 
Product Code:  SURV/171.E 
632 pp 
List Price:  $110.00 
MAA Member Price:  $99.00 
AMS Member Price:  $88.00 

Book DetailsMathematical Surveys and MonographsVolume: 171; 2011MSC: Primary 60; 15;
Random matrix theory is a wide and growing field with a variety of concepts, results, and techniques and a vast range of applications in mathematics and the related sciences. The book, written by wellknown experts, offers beginners a fairly balanced collection of basic facts and methods (Part 1 on classical ensembles) and presents experts with an exposition of recent advances in the subject (Parts 2 and 3 on invariant ensembles and ensembles with independent entries).
The text includes many of the authors' results and methods on several main aspects of the theory, thus allowing them to present a unique and personal perspective on the subject and to cover many topics using a unified approach essentially based on the Stieltjes transform and orthogonal polynomials. The exposition is supplemented by numerous comments, remarks, and problems. This results in a book that presents a detailed and selfcontained treatment of the basic random matrix ensembles and asymptotic regimes.
This book will be an important reference for researchers in a variety of areas of mathematics and mathematical physics. Various chapters of the book can be used for graduate courses; the main prerequisite is a basic knowledge of calculus, linear algebra, and probability theory.ReadershipGraduate students and research mathematicians interested in random matrix theory and its applications.

Table of Contents

Chapters

1. Introduction

Part 1. Classical ensembles

2. Gaussian ensembles: Semicircle law

3. Gaussian ensembles: Central Limit Theorem for linear eigenvalue statistics

4. Gaussian ensembles: Joint eigenvalue distribution and related results

5. Gaussian unitary ensemble

6. Gaussian orthogonal ensemble

7. Wishart and Laguerre ensembles

8. Classical compact groups ensembles: Global regime

9. Classical compact group ensembles: Further results

10. Law of addition of random matrices

Part 2. Matrix models

11. Matrix models: Global regime

12. Bulk universality for Hermitian matrix models

13. Universality for special points of Hermitian matrix models

14. Jacobi matrices and limiting laws for linear eigenvalue statistics

15. Universality for real symmetric matrix models

16. Unitary matrix models

Part 3. Ensembles with independent and weakly dependent entries

17. Matrices with Gaussian correlated entries

18. Wigner ensembles

19. Sample covariance and related matrices


Additional Material

Reviews

While a wide variety of ensembles are studied in this text, the methods are coherently focused, relying heavily in particular on Stieltjes transform based tools. This gives a slightly different perspective on the subject from other recent texts which often focus on other methods.
Mathematical Reviews


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Random matrix theory is a wide and growing field with a variety of concepts, results, and techniques and a vast range of applications in mathematics and the related sciences. The book, written by wellknown experts, offers beginners a fairly balanced collection of basic facts and methods (Part 1 on classical ensembles) and presents experts with an exposition of recent advances in the subject (Parts 2 and 3 on invariant ensembles and ensembles with independent entries).
The text includes many of the authors' results and methods on several main aspects of the theory, thus allowing them to present a unique and personal perspective on the subject and to cover many topics using a unified approach essentially based on the Stieltjes transform and orthogonal polynomials. The exposition is supplemented by numerous comments, remarks, and problems. This results in a book that presents a detailed and selfcontained treatment of the basic random matrix ensembles and asymptotic regimes.
This book will be an important reference for researchers in a variety of areas of mathematics and mathematical physics. Various chapters of the book can be used for graduate courses; the main prerequisite is a basic knowledge of calculus, linear algebra, and probability theory.
Graduate students and research mathematicians interested in random matrix theory and its applications.

Chapters

1. Introduction

Part 1. Classical ensembles

2. Gaussian ensembles: Semicircle law

3. Gaussian ensembles: Central Limit Theorem for linear eigenvalue statistics

4. Gaussian ensembles: Joint eigenvalue distribution and related results

5. Gaussian unitary ensemble

6. Gaussian orthogonal ensemble

7. Wishart and Laguerre ensembles

8. Classical compact groups ensembles: Global regime

9. Classical compact group ensembles: Further results

10. Law of addition of random matrices

Part 2. Matrix models

11. Matrix models: Global regime

12. Bulk universality for Hermitian matrix models

13. Universality for special points of Hermitian matrix models

14. Jacobi matrices and limiting laws for linear eigenvalue statistics

15. Universality for real symmetric matrix models

16. Unitary matrix models

Part 3. Ensembles with independent and weakly dependent entries

17. Matrices with Gaussian correlated entries

18. Wigner ensembles

19. Sample covariance and related matrices

While a wide variety of ensembles are studied in this text, the methods are coherently focused, relying heavily in particular on Stieltjes transform based tools. This gives a slightly different perspective on the subject from other recent texts which often focus on other methods.
Mathematical Reviews