Orthogonal Polynomials and Random Matrices: A Riemann-Hilbert Approach

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American Mathematical Soc., 1999 - 261 Seiten
This volume expands on a set of lectures held at the Courant Institute on Riemann-Hilbert problems, orthogonal polynomials, and random matrix theory. The goal of the course was to prove universality for a variety of statistical quantities arising in the theory of random matrix models. The central question was the following: Why do very general ensembles of random n times n matrices exhibit universal behavior as n > infinity? The main ingredient in the proof is the steepest descent method for oscillatory Riemann-Hilbert problems. Titles in this series are copublished with the Courant Institute of Mathematical Sciences at New York University.
 

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Inhalt

RiemannHilbert Problems
1
12 Examples
4
Jacobi Operators
13
22 The Spectrum of Jacobi Matrices
23
23 The Toda Flow
25
24 Unbounded Jacobi Operators
26
Support of a Measure
35
Orthogonal Polynomials
37
Universality
120
Equilibrium Measures
129
62 Existence of the Equilibrium Measure 𝜇ⱽ
134
63 Convergence of 𝛌ₓⁿ
145
64 Convergence of 1𝘯ℛ𝘹₁𝒅𝘹₁
149
65 Convergence of 𝜂ₓⁿ
159
66 Variational Problem for the Equilibrium Measure
167
67 Equilibrium Measure for 𝘝𝘹 𝘵𝘹²𝘮
169

32 A RiemannHilbert Problem
43
33 Some Symmetry Considerations
49
34 Zeros of Orthogonal Polynomials
52
Continued Fractions
57
42 Measure Theory and Ergodic Theory
64
43 Application to Jacobi Operators
76
44 Remarks on the Continued Fraction Expansion of a Number
85
Random Matrix Theory
89
52 Unitary Ensembles
91
53 Spectral Variables for Hermitian Matrices
94
54 Distribution of Eigenvalues
101
55 Distribution of Spacings of Eigenvalues
113
The Transfinite Diameter and Fekete Sets
179
Asymptotics for Orthogonal Polynomials
181
72 RiemannHilbert Problem for Orthogonal Polynomials
189
73 Deformation of a RiemannHilbert Problem
191
74 Asymptotics of Orthogonal Polynomials
201
75 Some Analytic Considerations of RiemannIlilbert Problems
208
76 Construction of the Parametrix
213
77 Asymptotics of Orthogonal Polynomials on the Real Axis
230
Universality
237
82 Asymptotics of Ps
251
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