Discrete Orthogonal Polynomials

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

• ISBN13: 9780691127347
• ISBN10: 0691127344
• Format: Paperback
• Copyright: 2007-01-02
• Publisher: Princeton Univ Pr

Summary

This book describes the theory and applications of discrete orthogonal polynomials--polynomials that are orthogonal on a finite set. Unlike other books,Discrete Orthogonal Polynomialsaddresses 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.

Author Biography

J. Baik is Associate Professor of Mathematics at the University of Michigan. T. Kriecherbauer is Professor of Mathematics at Ruhr-Universitat 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	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
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