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Skew-Orthogonal Polynomials and Random Matrix Theory

Saugata Ghosh , Gurgaon, India
A co-publication of the AMS and Centre de Recherches Mathématiques
Available Formats:
Hardcover ISBN: 978-0-8218-4878-4
Product Code: CRMM/28
List Price: $58.00 MAA Member Price:$52.20
AMS Member Price: $46.40 Electronic ISBN: 978-1-4704-1771-0 Product Code: CRMM/28.E List Price:$54.00
MAA Member Price: $48.60 AMS Member Price:$43.20
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This product is available for purchase as a bundle. Purchasing as a bundle enables you to save on the electronic version.
List Price: $87.00 MAA Member Price:$78.30
AMS Member Price: $69.60 Click above image for expanded view Skew-Orthogonal Polynomials and Random Matrix Theory Saugata Ghosh , Gurgaon, India A co-publication of the AMS and Centre de Recherches Mathématiques Available Formats:  Hardcover ISBN: 978-0-8218-4878-4 Product Code: CRMM/28  List Price:$58.00 MAA Member Price: $52.20 AMS Member Price:$46.40
 Electronic ISBN: 978-1-4704-1771-0 Product Code: CRMM/28.E
 List Price: $54.00 MAA Member Price:$48.60 AMS Member Price: $43.20 Bundle Print and Electronic Formats and Save! This product is available for purchase as a bundle. Purchasing as a bundle enables you to save on the electronic version.  List Price:$87.00 MAA Member Price: $78.30 AMS Member Price:$69.60
• Book Details

CRM Monograph Series
Volume: 282009; 127 pp
MSC: Primary 33; 11; 26; 15;

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.

Research mathematicians interested in random matrix theory.

• Chapters
• Introduction
• Level density and correlation functions
• The $S^{(\beta )}_\mathbb {N}(x,y)$ kernel and Christoffel–Darboux formulas
• Mapping
• Unitary ensembles
• Orthogonal ensembles (even dimension)
• Orthogonal ensembles (odd dimension)
• Symplectic ensembles
• Skew-orthogonal polynomials and differential systems
• Matrix integral representations and zeros of polynomials
• Duality
• Conclusion

• Requests

Review Copy – for reviewers who would like to review an AMS book
Accessibility – to request an alternate format of an AMS title
Volume: 282009; 127 pp
MSC: Primary 33; 11; 26; 15;

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.

Research mathematicians interested in random matrix theory.

• Chapters
• Introduction
• Level density and correlation functions
• The $S^{(\beta )}_\mathbb {N}(x,y)$ kernel and Christoffel–Darboux formulas
• Mapping
• Unitary ensembles
• Orthogonal ensembles (even dimension)
• Orthogonal ensembles (odd dimension)
• Symplectic ensembles
• Skew-orthogonal polynomials and differential systems
• Matrix integral representations and zeros of polynomials
• Duality
• Conclusion
Review Copy – for reviewers who would like to review an AMS book
Accessibility – to request an alternate format of an AMS title
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