Orthogonal polynomials and random matrices
Subtitle: a Riemann-Hilbert approach
By: Percy Deift.
Series: Courant lecture notes ;; 3
Publisher: American Mathematical Society
Published: 2000
Place of publication: Providence, R.I
Language: eng
Number of pages: 261
Subjects: Orthogonal polynomials; Random matrices
Notes: Includes bibliographical references (p. 259-261).
Originally published: New York : Courant Institute of Mathematical Sciences, New York University, c1999.
Table of contents: title: Machine generated contents note: Chapter 1. Riemann-Hilbert Problems 1; title: 1.1. What Is a Riemann-Hilbert Problem? 1; title: 1.2. Examples 4; title: Chapter 2. Jacobi Operators 13; title: 2.1. Jacobi Matrices 13; title: 2.2. The Spectrum of Jacobi Matrices 23; title: 2.3. The Toda Flow 25; title: 2.4. Unbounded Jacobi Operators 26; title: 2.5. Appendix: Support of a Measure 35; title: Chapter 3. Orthogonal Polynomials 37; title: 3.1. Construction of Orthogonal Polynomials 37; title: 3.2. A Riemann-Hilbert Problem 43; title: 3.3. Some Symmetry Considerations 49; title: 3.4. Zeros of Orthogonal Polynomials 52; title: Chapter 4. Continued Fractions 57; title: 4.1. Continued Fraction Expansion of a Number 57; title: 4.2. Measure Theory and Ergodic Theory 64; title: 4.3. Application to Jacobi Operators 76; title: 4.4. Remarks on the Continued Fraction Expansion of a Number 85; title: Chapter 5. Random Matrix Theory 89; title: 5.1. Introduction 89; title: 5.2. Unitary Ensembles 91; title: 5.3. Spectral Variables for Hermitian Matrices 94; title: 5.4. Distribution of Eigenvalues 101; title: 5.5. Distribution of Spacings of Eigenvalues 113; title: 5.6. Further Remarks on the Nearest-Neighbor Spacing Distribution and; title: Universality 120; title: Chapter 6. Equilibrium Measures 129; title: 6.1. Scaling 129; title: 6.2. Existence of the Equilibrium Measure LLV 134; title: 6.3. Convergence of X,* 145; title: 6.4. Convergence of RlI(xl)dxl 149; title: 6.5. Convergence of rlx* 159; title: 6.6. Variational Problem for the Equilibrium Measure 167; title: 6.7. Equilibrium Measure for V(x) = tx2m 169; title: 6.8. Appendix: The Transfinite Diameter and Fekete Sets 179; title: Chapter 7. Asymptotics for Orthogonal Polynomials 181; title: 7.1. Riemann-Hilbert Problem: The Precise Sense 181; title: 7.2. Riemann-Hilbert Problem for Orthogonal Polynomials 189; title: 7.3. Deformation of a Riemann-Hilbert Problem 191; title: 7.4. Asymptotics of Orthogonal Polynomials 201; title: 7.5. Some Analytic Considerations of Riemann-Hilbert Problems 208; title: 7.6. Construction of the Parametrix 213; title: 7.7. Asymptotics of Orthogonal Polynomials on the Real Axis 230; title: Chapter 8. Universality 237; title: 8.1. Universality 237; title: 8.2. Asymptotics of Ps 251.