Orthogonal Polynomials and Random Matrices: A Riemann-Hilbert ApproachThis 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
| 1 | |
| 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 |
Andere Ausgaben - Alle anzeigen
Orthogonal Polynomials and Random Matrices: A Riemann-Hilbert Approach: A ... Percy Deift Eingeschränkte Leseprobe - 2000 |
Häufige Begriffe und Wortgruppen
analytic associated assume asymptotics bounded calculations Chapter choose clearly compact support compute conclude consider constant continued fraction contour converges define denote described distribution du(s du(x eigenvalues equation equilibrium measure example exercise exists expect fact Figure finite fixed follows formula function given hand Hence implies integral interest interval leading least Lemma limit matrix mean measure neighborhood Note Observe obtain operator orthogonal polynomials particular positive precisely Prob probability measure proof prove R-H problem Recall relation REMARK require respect result Riemann-Hilbert roots satisfies scaled self-adjoint sense side solution solves space spectral sufficiently Suppose symmetric theorem theory true unique universality values zero
Verweise auf dieses Buch
Semiclassical Soliton Ensembles for the Focusing Nonlinear Schrodinger ... Spyridon Kamvissis,Kenneth D.T-R McLaughlin,Peter D. Miller Keine Leseprobe verfügbar - 2003 |
XIVth International Congress on Mathematical Physics: Lisbon, 28 July - 2 ... Jean-Claude Zambrini Eingeschränkte Leseprobe - 2005 |
Bibliografische Informationen
| Titel | Orthogonal Polynomials and Random Matrices: A Riemann-Hilbert Approach Band 3 von Courant lecture notes in mathematics Band 3 von Courant lecture notes in mathematics: Courant Institute of Mathematical Sciences Ausgabe 3 von Courant lecture notes |
| Autor/in | Percy Deift |
| Verlag | American Mathematical Soc., 1999 |
| ISBN | 0821883445, 9780821883440 |
| Länge | 261 Seiten |
| Zitat exportieren | BiBTeX EndNote RefMan |