Orthogonal polynomials and random matrices a Riemann-Hilbert approach Percy Deift

Originally published: New York : Courant Institute of Mathematical Sciences, New York University, c1999

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