Discrete Orthogonal Polynomials. (AM-164)

Asymptotics and Applications (AM-164)

By: J. Baik, T. Kriecherbauer, Kenneth Dean T-R McLaughlin, Peter D. Miller

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Paperback | 2 April 2007

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184 Pages

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25.4 x 20.3 x 1.27

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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.

Industry 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

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

Discrete Orthogonal Polynomials. (AM-164)

Asymptotics and Applications (AM-164)

At a Glance

Paperback

Industry Reviews

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