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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
Skew-Orthogonal Polynomials and Random Matrix Theory
Hardcover ISBN:  978-0-8218-4878-4
Product Code:  CRMM/28
List Price: $115.00
MAA Member Price: $103.50
AMS Member Price: $92.00
eBook ISBN:  978-1-4704-1771-0
Product Code:  CRMM/28.E
List Price: $110.00
MAA Member Price: $99.00
AMS Member Price: $88.00
Hardcover ISBN:  978-0-8218-4878-4
eBook: ISBN:  978-1-4704-1771-0
Product Code:  CRMM/28.B
List Price: $225.00 $170.00
MAA Member Price: $202.50 $153.00
AMS Member Price: $180.00 $136.00
Skew-Orthogonal Polynomials and Random Matrix Theory
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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
Hardcover ISBN:  978-0-8218-4878-4
Product Code:  CRMM/28
List Price: $115.00
MAA Member Price: $103.50
AMS Member Price: $92.00
eBook ISBN:  978-1-4704-1771-0
Product Code:  CRMM/28.E
List Price: $110.00
MAA Member Price: $99.00
AMS Member Price: $88.00
Hardcover ISBN:  978-0-8218-4878-4
eBook ISBN:  978-1-4704-1771-0
Product Code:  CRMM/28.B
List Price: $225.00 $170.00
MAA Member Price: $202.50 $153.00
AMS Member Price: $180.00 $136.00
  • 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.

    Titles in this series are co-published with the Centre de recherches mathématiques.

    Readership

    Research mathematicians interested in random matrix theory.

  • Table of Contents
     
     
    • 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 publishers of book reviews
    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.

Titles in this series are co-published with the Centre de recherches mathématiques.

Readership

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 publishers of book reviews
Accessibility – to request an alternate format of an AMS title
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