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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 |
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