**CRM Monograph Series**

Volume: 28;
2009;
127 pp;
Hardcover

MSC: Primary 33; 11; 26; 15;

Print ISBN: 978-0-8218-4878-4

Product Code: CRMM/28

List Price: $54.00

Individual Member Price: $43.20

**Electronic ISBN: 978-1-4704-1771-0
Product Code: CRMM/28.E**

List Price: $54.00

Individual Member Price: $43.20

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#### Supplemental Materials

# Skew-Orthogonal Polynomials and Random Matrix Theory

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*Saugata Ghosh*

A co-publication of the AMS and Centre de Recherches Mathématiques

Orthogonal polynomials satisfy a three-term recursion relation
irrespective of the weight function with respect to which they are
defined. This gives a simple formula for the kernel function, known in
the literature as the Christoffel–Darboux sum. The availability
of asymptotic results of orthogonal polynomials and the simple
structure of the Christoffel–Darboux sum make the study of
unitary ensembles of random matrices relatively straightforward.

In this book, the author develops the theory of skew-orthogonal
polynomials and obtains recursion relations which, unlike orthogonal
polynomials, depend on weight functions. After deriving reduced
expressions, called the generalized Christoffel–Darboux formulas
(GCD), he obtains universal correlation functions and non-universal
level densities for a wide class of random matrix ensembles using the
GCD.

The author also shows that once questions about higher order
effects are considered (questions that are relevant in different
branches of physics and mathematics) the use of the GCD promises to be
efficient.

Titles in this series are co-published with the Centre de Recherches Mathématiques.

#### Readership

Research mathematicians interested in random matrix theory.

#### Table of Contents

# Table of Contents

## Skew-Orthogonal Polynomials and Random Matrix Theory

- Cover Cover11
- Title page i2
- Contents v6
- Acknowledgments vii8
- Introduction 110
- Level density and correlation functions 1120
- The 𝑆^{(𝛽)}_{}ℕ(𝕩,𝕪) kernel and Christoffel–Darboux formulas 2534
- Mapping 3544
- Unitary ensembles 4554
- Orthogonal ensembles (even dimension) 5160
- Orthogonal ensembles (odd dimension) 6372
- Symplectic ensembles 6574
- Skew-orthogonal polynomials and differential systems 7988
- Matrix integral representations and zeros of polynomials 105114
- Duality 107116
- Conclusion 113122
- Appendix A. Proofs of (5.7), (5.12), and (5.19) 115124
- Appendix B. Associated Laguerre and Gaussian results as limiting cases of Jacobi skew-orthogonal polynomials 119128
- Appendix C. Proofs of (10.2)–(10.9) 121130
- Bibliography 125134
- Back Cover Back Cover1138