Orthogonal polynomials and random matrices: A Riemann-Hilbert approach (Courant lecture notes in mathematics)

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 $n {\rightarrow} {\infty}$? The main ingredient in the proof is the steepest descent method for oscillatory Riemann-Hilbert problems. Machine generated contents note: Chapter 1. Riemann-Hilbert Problems 1 1.1. What Is a Riemann-Hilbert Problem? 1 1.2. Examples 4 Chapter 2. Jacobi Operators 13 2.1. Jacobi Matrices 13 2.2. The Spectrum of Jacobi Matrices 23 2.3. The Toda Flow 25 2.4. Unbounded Jacobi Operators 26 2.5. Appendix: Support of a Measure 35 Chapter 3. Orthogonal Polynomials 37 3.1. Construction of Orthogonal Polynomials 37 3.2. A Riemann-Hilbert Problem 43 3.3. Some Symmetry Considerations 49 3.4. Zeros of Orthogonal Polynomials 52 Chapter 4. Continued Fractions 57 4.1. Continued Fraction Expansion of a Number 57 4.2. Measure Theory and Ergodic Theory 64 4.3. Application to Jacobi Operators 76 4.4. Remarks on the Continued Fraction Expansion of a Number 85 Chapter 5. Random Matrix Theory 89 5.1. Introduction 89 5.2. Unitary Ensembles 91 5.3. Spectral Variables for Hermitian Matrices 94 5.4. Distribution of Eigenvalues 101 5.5. Distribution of Spacings of Eigenvalues 113 5.6. Further Remarks on the Nearest-Neighbor Spacing Distribution and Universality 120 Chapter 6. Equilibrium Measures 129 6.1. Scaling 129 6.2. Existence of the Equilibrium Measure LLV 134 6.3. Convergence of X,* 145 6.4. Convergence of RlI(xl)dxl 149 6.5. Convergence of rlx* 159 6.6. Variational Problem for the Equilibrium Measure 167 6.7. Equilibrium Measure for V(x) = tx2m 169 6.8. Appendix: The Transfinite Diameter and Fekete Sets 179 Chapter 7. Asymptotics for Orthogonal Polynomials 181 7.1. Riemann-Hilbert Problem: The Precise Sense 181 7.2. Riemann-Hilbert Problem for Orthogonal Polynomials 189 7.3. Deformation of a Riemann-Hilbert Problem 191 7.4. Asymptotics of Orthogonal Polynomials 201 7.5. Some Analytic Considerations of Riemann-Hilbert Problems 208 7.6. Construction of the Parametrix 213 7.7. Asymptotics of Orthogonal Polynomials on the Real Axis 230 Chapter 8. Universality 237 8.1. Universality 237 8.2. Asymptotics of Ps 251.