Hardcover ISBN:  9780821848784 
Product Code:  CRMM/28 
List Price:  $58.00 
MAA Member Price:  $52.20 
AMS Member Price:  $46.40 
Electronic ISBN:  9781470417710 
Product Code:  CRMM/28.E 
List Price:  $54.00 
MAA Member Price:  $48.60 
AMS Member Price:  $43.20 

Book DetailsCRM Monograph SeriesVolume: 28; 2009; 127 ppMSC: Primary 33; 11; 26; 15;
Orthogonal polynomials satisfy a threeterm 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 skeworthogonal 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 nonuniversal 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.ReadershipResearch 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

Skeworthogonal polynomials and differential systems

Matrix integral representations and zeros of polynomials

Duality

Conclusion


Additional Material

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Orthogonal polynomials satisfy a threeterm 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 skeworthogonal 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 nonuniversal 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

Skeworthogonal polynomials and differential systems

Matrix integral representations and zeros of polynomials

Duality

Conclusion