Discrete Orthogonal Polynomials. (AM-164): Asymptotics and Applications (AM-164)

✍ Scribed by J. Baik; T. Kriecherbauer; Kenneth D.T-R McLaughlin; Peter D. Miller

• Publisher: Princeton University Press
• Year: 2007
• Tongue: English
• Leaves: 178
• Series: Annals of Mathematics Studies; 164
• Edition: Course Book
• Category: Library

✦ Synopsis

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.

✦ Table of Contents

Contents
Preface
Chapter 1. Introduction
1.1 Motivating applications
1.2 Discrete orthogonal polynomials
1.3 Assumptions
1.4 Goals and methodology
1.5 Outline of the rest of the book
1.6 Research background
Chapter 2. Asymptotics of General Discrete Orthogonal Polynomials in the Complex Plane
2.1 The equilibrium energy problem
2.2 Elements of hyperelliptic function theory
2.3 Results on asymptotics of discrete orthogonal polynomials
2.4 Equilibrium measures for some classical discrete orthogonal polynomials
Chapter 3. Applications
3.1 Discrete orthogonal polynomial ensembles and their particle statistics
3.2 Dual ensembles and hole statistics
3.3 Results on asymptotic universality for general weights
3.4 Random rhombus tilings of a hexagon
3.5 The continuum limit of the Toda lattice
Chapter 4. An Equivalent Riemann-Hilbert Problem
4.1 Choice of Δ: the transformation from P(z; N, k) to Q(z; N, k)
4.2 Removal of poles in favor of discontinuities along contours: the transformation from Q(z; N, k) to R(z)
4.3 Use of the equilibrium measure: the transformation from R(z) to S(z)
4.4 Steepest descent: the transformation from S(z) to X(z)
4.5 Properties of X(z)
Chapter 5. Asymptotic Analysis
5.1 Construction of a global parametrix for X(z)
5.2 Error estimation
Chapter 6. Discrete Orthogonal Polynomials: Proofs of Theorems Stated in §2.3
6.1 Asymptotic analysis of P(z; N, k) for z Є C \ [a, b]
6.2 Asymptotic behavior of πN,k(z) for z near a void of [a, b]: the proof of Theorem 2.9
6.3 Asymptotic behavior of πN,k(z) for z near a saturated region of [a, b]
6.4 Asymptotic behavior of πN,k(z) for z near a band
6.5 Asymptotic behavior of πN,k(z) for z near a band edge
Chapter 7. Universality: Proofs of Theorems Stated in §3.3
7.1 Relation between correlation functions of dual ensembles
7.2 Exact formulae for KN,k(x, y)
7.3 Asymptotic formulae for KN,k(x, y) and universality
Appendix A. The Explicit Solution of Riemann-Hilbert Problem 5.1
A.1 Steps for making the jump matrix piecewise-constant: the transformation from X(z) to Y#(z)
A.2 Construction of Y#(z) using hyperelliptic function theory
A.3 The matrix X(z) and its properties
Appendix B. Construction of the Hahn Equilibrium Measure: the Proof of Theorem 2.17
B.1 General strategy: the one-band ansatz
B.2 The void-band-void configuration
B.3 The saturated-band-void configuration
B.4 The void-band-saturated configuration
B.5 The saturated-band-saturated configuration
Appendix C. List of Important Symbols
Bibliography
Index