Discrete Orthogonal Polynomials: Asymptotics and Applications

by J. Baik, T. Kriecherbauer, Kenneth D.T-R McLaughlin, Peter D. Miller

Overview

This book describes the theory and applications of discrete orthogonal polynomials—polynomials that are orthogonal on a finite set. Unlike other books, Discrete Orthogonal Polynomials addresses completely general weight functions and presents a new methodology for handling the discrete weights case.

J. Baik, T. Kriecherbauer, K. T.-R. McLaughlin & P. D. Miller focus on asymptotic aspects of general, nonclassical discrete orthogonal polynomials and set out applications of current interest. Topics covered include the probability theory of discrete orthogonal polynomial ensembles and the continuum limit of the Toda lattice. The primary concern throughout is the asymptotic behavior of discrete orthogonal polynomials for general, nonclassical measures, in the joint limit where the degree increases as some fraction of the total number of points of collocation. The book formulates the orthogonality conditions defining these polynomials as a kind of Riemann-Hilbert problem and then generalizes the steepest descent method for such a problem to carry out the necessary asymptotic analysis.

Product Details

ISBN-13:	9780691127347
Publisher:	Princeton University Press
Publication date:	01/22/2007
Series:	Annals of Mathematics Studies , #164
Pages:	184
Product dimensions:	8.00(w) x 10.00(h) x (d)

About the Author

J. Baik is Associate Professor of Mathematics at the University of Michigan. T. Kriecherbauer is Professor of Mathematics at Ruhr-Universität Bochum in Bochum, Germany. K. T.-R. McLaughlin is Professor of Mathematics at the University of Arizona. P. D. Miller is Associate Professor of Mathematics at the University of Michigan.

Table of Contents

Preface     vii
Introduction     1
Motivating applications     1
Discrete orthogonal polynomials     8
Assumptions     10
Goals and methodology     11
Outline of the rest of the book     22
Research background     23
Asymptotics of General Discrete Orthogonal Polynomials in the Complex Plane     25
The equilibrium energy problem     25
Elements of hyperelliptic function theory     31
Results on asymptotics of discrete orthogonal polynomials     33
Equilibrium measures for some classical discrete orthogonal polynomials     41
Applications     49
Discrete orthogonal polynomial ensembles and their particle statistics     49
Dual ensembles and hole statistics     51
Results on asymptotic universality for general weights     52
Random rhombus tilings of a hexagon     57
The continuum limit of the Toda lattice     60
An Equivalent Riemann-Hilbert Problem     67
Choice of [Delta]: the transformation from P(z; N, k) to Q(z; N, k)     67
Removal of poles in favor of discontinuities along contours: the transformation from Q(z; N, k) to R(z)     69
Use of the equilibrium measure: thetransformation from R(z) to S(z)     70
Steepest descent: the transformation from S(z) to X(z)     78
Properties of X(z)     79
Asymptotic Analysis     87
Construction of a global parametrix for X(z)     87
Error estimation     99
Discrete Orthogonal Polynomials: Proofs of Theorems Stated in [Sect]2.3     105
Asymptotic analysis of P(z; N, k) for z [isin] C \ [a, b]     105
Asymptotic behavior of [pi subscript N,k] (z) for z near a void of [a, b]: the proof of Theorem 2.9     107
Asymptotic behavior of [pi subscript N,k] (z) for z near a saturated region of [a, b]     108
Asymptotic behavior of [pi subscript N,k] (z) for z near a band     110
Asymptotic behavior of [pi subscript N,k] (z) for z near a band edge     112
Universality: Proofs of Theorems Stated in [Sect]3.3     115
Relation between correlation functions of dual ensembles     115
Exact formulae for K [subscript N, k] (x, y)     118
Asymptotic formulae for K [subscript N, k] (x, y) and universality     124
The Explicit Solution of Riemann-Hilbert Problem 5.1     135
Steps for making the jump matrix piecewise-constant: the transformation from X(z) to Y[superscript #](z)     135
Construction of Y[superscript #](z) using hyperelliptic function theory     137
The matrix X(z) and its properties     141
Construction of the Hahn Equilibrium Measure: the Proof of Theorem 2.17     145
General strategy: the one-band ansatz     145
The void-band-void configuration     146
The saturated-band-void configuration     149
The void-band-saturated configuration     150
The saturated-band-saturated configuration     151
List of Important Symbols     153
Bibliography     163
Index     167

Editorial Reviews

"This volume is a well written contribution to the theory of asymptotics of discrete orthogonal polynomials and its applications."—-H.J. Glaeske, Zentralblatt MATH

"The book includes the authors' own research results developed over the last years. Their approach and proofs are straightforward constructive making this monograph accessible and valuable to undergraduate and graduate students, PhD students and researchers involved in the asymptotic analysis of systems of discrete orthogonal polynomials."—-Octavian Agratini, Mathematica

"This book is very interesting. It pulls together several methods developed during the last twenty years and makes them into a general theory. Thanks to this general theory, several open problems are solved. The organization of the book is very nice."—-Sylvie Corteel, Mathematical Reviews

From the Publisher

This book is very interesting. It pulls together several methods developed during the last twenty years and makes them into a general theory. Thanks to this general theory, several open problems are solved. The organization of the book is very nice.

Mathematical Reviews - Sylvie Corteel

The book includes the authors' own research results developed over the last years. Their approach and proofs are straightforward constructive making this monograph accessible and valuable to undergraduate and graduate students, PhD students and researchers involved in the asymptotic analysis of systems of discrete orthogonal polynomials.

Mathematica - Octavian Agratini

This volume is a well written contribution to the theory of asymptotics of discrete orthogonal polynomials and its applications.

Zentralblatt MATH - H.J. Glaeske

This volume is a well written contribution to the theory of asymptotics of discrete orthogonal polynomials and its applications.
— H.J. Glaeske

Zentralblatt MATH Database

The book includes the authors' own research results developed over the last years. Their approach and proofs are straightforward constructive making this monograph accessible and valuable to undergraduate and graduate students, PhD students and researchers involved in the asymptotic analysis of systems of discrete orthogonal polynomials.
— Octavian Agratini

Mathematica

This volume is a well written contribution to the theory of asymptotics of discrete orthogonal polynomials and its applications.
— H.J. Glaeske

Zentralblatt MATH

This book is very interesting. It pulls together several methods developed during the last twenty years and makes them into a general theory. Thanks to this general theory, several open problems are solved. The organization of the book is very nice.
— Sylvie Corteel

Mathematical Reviews