Discrete orthogonal polynomials : asymptotics and applications

✍ Scribed by Jinho Baik; et al

• Publisher: Princeton University Press
• Year: 2007
• Tongue: English
• Leaves: 172
• Series: Annals of mathematics studies, no. 164
• 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."--BOOK JACKET. Read more... Asymptotics of General Discrete Orthogonal Polynomials in the Complex Plane -- Applications -- An Equivalent Riemann-Hilbert Problem -- Asymptotic Analysis -- Discrete Orthogonal Polynomials: Proofs of Theorems Stated in 2.3 -- Universality: Proofs of Theorems Stated in 3.3

✦ Table of Contents

Cover......Page 1
Copyright......Page 2
Contents......Page 3
Preface......Page 5
1.1.1 Discrete orthogonal polynomial ensembles......Page 6
1.1.2 The continuum limit of the Toda lattice......Page 9
1.2 Discrete orthogonal polynomials......Page 13
1.3.1 Basic assumptions......Page 15
1.4 Goals and methodology......Page 16
1.4.1 The basic interpolation problem......Page 17
1.4.2 Exponentially deformed weights and the Toda lattice......Page 19
1.4.3 Triangularity of residue matrices and dual polynomials......Page 22
1.4.4 Overview of the key steps......Page 24
1.5 Outline of the rest of the book......Page 27
1.6 Research background......Page 28
2.1.1 The equilibrium measure......Page 29
2.1.2 Simplifying assumptions on the equilibrium measure......Page 30
2.1.4 Quantities derived from the equilibrium measure......Page 32
2.1.5 Dual equilibrium measures......Page 34
2.2 Elements of hyperelliptic function theory......Page 35
2.3 Results on asymptotics of discrete orthogonal polynomials......Page 37
2.4 Equilibrium measures for some classical discrete orthogonal polynomials......Page 45
2.4.1 The Krawtchouk polynomials......Page 46
2.4.2 The Hahn and associated Hahn polynomials......Page 47
3.1 Discrete orthogonal polynomial ensembles and their particle statistics......Page 53
3.2 Dual ensembles and hole statistics......Page 55
3.3 Results on asymptotic universality for general weights......Page 56
3.4.1 Relation to the Hahn and associated Hahn ensembles......Page 61
3.4.2 Statistical asymptotics......Page 63
3.5.1 Solution procedure......Page 64
3.5.3 Defect motion in saturated regions......Page 66
3.5.4 Asymptotics of the linearized problem......Page 67
4.1 Choice: the transformation from P(z; N, k) to Q(z; N, k)......Page 71
4.2 Removal of poles in favor of discontinuities along contours: the transformation from Q(z; N, k) to R(z)......Page 73
4.3.2 The jump of S(z) on the real axis......Page 74
4.3.3 Important properties of the functions T(z) and T(z)......Page 76
4.4 Steepest descent: the transformation from S(z) to X(z)......Page 82
4.5 Properties of X(z)......Page 83
5.1.1 Outer asymptotics......Page 91
5.1.2 Inner asymptotics near band edges......Page 93
5.2 Error estimation......Page 103
6.1.1 Asymptotic behavior: the proof of Theorem 2.7......Page 109
6.1.2 Asymptotic behavior of the leading coefficients and of the recurrence coefficients: the proof of Theorem 2.8......Page 110
6.2 Asymptotic behavior: the proof of Theorem 2.9......Page 111
6.3.1 Asymptotics valid away from hard edges: the proof of Theorem 2.10......Page 112
6.3.3 Asymptotics of the zeros in saturated regions: the proof of Theorem 2.12......Page 113
6.4.1 Asymptotic behavior: the proof of Theorem 2.13......Page 114
6.5.1 Band/void edges: the proof of Theorem 2.15......Page 116
6.5.2 Band/saturated region edges: the proof of Theorem 2.16......Page 117
7.1.1 Probabilistic approach......Page 119
7.1.2 Direct approach......Page 120
7.2 Exact formulae......Page 122
7.3 Asymptotic formulae and universality......Page 128
7.3.1 Asymptotic universality of statistics for particles in a band: the proofs of Theorems 3.1 and 3.2......Page 129
7.3.2 Correlation functions for particles in voids: the proofs of Theorems 3.3 and 3.4......Page 131
7.3.3 Correlation functions for particles in saturated regions: the proofs of Theorems 3.5 and 3.6......Page 132
7.3.4 Asymptotic universality of statistics for particles near band/gap edges: the proofs of Theorems 3.7, 3.8, 3.9, and 3.10......Page 133
A.1 Steps for making the jump matrix piecewise-constant: the transformation......Page 138
A.2 Construction using hyperelliptic function theory......Page 140
A.3 The matrix X(z) and its properties......Page 144
A.3.2 Completion of the proof of Proposition 5.3......Page 146
B.1 General strategy: the one-band ansatz......Page 148
B.2 The void-band-void configuration......Page 149
B.3 The saturated-band-void configuration......Page 152
B.4 The void-band-saturated configuration......Page 153
B.5 The saturated-band-saturated configuration......Page 154
Appendix C. List of Important Symbols......Page 155
Bibliography......Page 165
Index......Page 169