An introduction to random matrices:
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| Main Authors: | , , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Cambridge ; New York ; Melbourne ; Madrid ; Cape Town ; SIngapore ; Sao Paulo ; Delhi
Cambridge University Press
2010
|
| Edition: | first published |
| Series: | Cambridge studies in advanced mathematics
118 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | xiv, 492 Seiten |
| ISBN: | 9780521194525 |
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| 245 | 1 | 0 | |a An introduction to random matrices |c Greg W. Anderson ; Alice Guionnet ; Ofer Zeitouni |
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Record in the Search Index
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| adam_text | Titel: An introduction to random matrices
Autor: Anderson, Greg W.
Jahr: 2010
Contents
Preface page xiii
1 Introduction 1
2 Real and complex Wigner matrices 6
2.1 Real Wigner matrices: traces, moments and combinatorics 6
2.1.1 The semicircle distribution, Catalan numbers and
Dyck paths 7
2.1.2 Proof #1 of Wigner s Theorem 2.1.1 10
2.1.3 Proof of Lemma 2.1.6: words and graphs 11
2.1.4 Proof of Lemma 2.1.7: sentences and graphs 17
2.1.5 Some useful approximations 21
2.1.6 Maximal eigenvalues and Füredi-Komlós enumeration 23
2.1.7 Central limit theorems for moments 29
2.2 Complex Wigner matrices 35
2.3 Concentration for functionals of random matrices and
logarithmic Sobolev inequalities 38
2.3.1 Smoothness properties of linear functions of the
empirical measure 38
2.3.2 Concentration inequalities for independent variables
satisfying logarithmic Sobolev inequalities 39
2.3.3 Concentration for Wigner-type matrices 42
2.4 Stieltjes transforms and recursions 43
Contents
2.4.1 Gaussian Wigner matrices 45
2.4.2 General Wigner matrices 47
2.5 Joint distribution of eigenvalues in the GOE and the GUE 50
2.5.1 Definition and preliminary discussion of the GOE
and the GUE 51
2.5.2 Proof of the joint distribution of eigenvalues 54
2.5.3 Selberg s integral formula and proof of (2.5.4) 58
2.5.4 Joint distribution of eigenvalues: alternative formu-
lation 65
2.5.5 Superposition and decimation relations 66
2.6 Large deviations for random matrices 70
2.6.1 Large deviations for the empirical measure 71
2.6.2 Large deviations for the top eigenvalue 81
2.7 Bibliographical notes 85
Hermit e polynomials, spacings and limit distributions for the Gaus-
sian ensembles 90
3.1 Summary of main results: spacing distributions in the bulk
and edge of the spectrum for the Gaussian ensembles 90
3.1.1 Limit results for the GUE 90
3.1.2 Generalizations: limit formulas for the GOE and GSE 93
3.2 Hermite polynomials and the GUE 94
3.2.1 The GUE and determinantal laws 94
3.2.2 Properties of the Hermite polynomials and oscillator
wave-functions 99
3.3 The semicircle law revisited 101
3.3.1 Calculation of moments of ¿at 102
3.3.2 The Harer-Zagier recursion and Ledoux s argument 103
3.4 Quick introduction to Fredholm determinants 107
3.4.1 The setting, fundamental estimates and definition of
the Fredholm determinant 107
3.4.2 Definition of the Fredholm adjugant, Fredholm
resolvent and a fundamental identity 110
Contents ix
3.5 Gap probabilities atO and proof of Theorem 3.1.1 114
3.5.1 The method of Laplace 115
3.5.2 Evaluation of the scaling limit: proof of Lemma
3.5.1 117
3.5.3 A complement: determinantal relations 120
3.6 Analysis of the sine-kernel 121
3.6.1 General differentiation formulas 121
3.6.2 Derivation of the differential equations: proof of
Theorem 3.6.1 126
3.6.3 Reduction to Painlevé V 128
3.7 Edge-scaling: proof of Theorem 3.1.4 132
3.7.1 Vague convergence of the largest eigenvalue: proof
of Theorem 3.1.4 133
3.7.2 Steepest descent: proof of Lemma 3.7.2 134
3.7.3 Properties of the Airy functions and proof of Lemma
3.7.1 139
3.8 Analysis of the Tracy-Widom distribution and proof of
Theorem 3.1.5 142
3.8.1 The first standard moves of the game 144
3.8.2 The wrinkle in the carpet 144
3.8.3 Linkage to Painlevé II 146
3.9 Limiting behavior of the GOE and the GSE 148
3.9.1 Pfaffians and gap probabilities 148
3.9.2 Fredholm representation of gap probabilities 155
3.9.3 Limit calculations 160
3.9.4 Differential equations 170
3.10 Bibliographical notes 181
Some generalities 186
4.1 Joint distribution of eigenvalues in the classical matrix
ensembles 187
4.1.1 Integration formulas for classical ensembles 187
4.1.2 Manifolds, volume measures and the coarea formula 193
Contents
4.1.3 An integration formula of Weyl type 199
4.1.4 Applications of Weyl s formula 206
4.2 Determinantal point processes 214
4.2.1 Point processes: basic definitions 215
4.2.2 Determinantal processes 220
4.2.3 Determinantal projections 222
4.2.4 The CLT for determinantal processes 227
4.2.5 Determinantal processes associated with eigenvalues 228
4.2.6 Translation invariant determinantal processes 232
4.2.7 One-dimensional translation invariant determinantal
processes 237
4.2.8 Convergence issues 241
4.2.9 Examples 243
4.3 Stochastic analysis for random matrices 248
4.3.1 Dyson s Brownian motion 249
4.3.2 A dynamical version of Wigner s Theorem 262
4.3.3 Dynamical central limit theorems 273
4.3.4 Large deviation bounds 277
4.4 Concentration of measure and random matrices 281
4.4.1 Concentration inequalities for Hermitian matrices
with independent entries 282
4.4.2 Concentration inequalities for matrices with depen-
dent entries 287
4.5 Tridiagonal matrix models and the ß ensembles 302
4.5.1 Tridiagonal representation of ß ensembles 303
4.5.2 Scaling limits at the edge of the spectrum 306
4.6 Bibliographical notes 318
Free probability 322
5.1 Introduction and main results 323
5.2 Noncommutative laws and noncommutative probability spaces 325
Contents xi
5.2.1 Algebraic noncommutative probability spaces and
laws 325
5.2.2 C*-probability spaces and the weak*-topology 329
5.2.3 W*-probability spaces 339
5.3 Free independence 348
5.3.1 Independence and free independence 348
5.3.2 Free independence and combinatorics 354
5.3.3 Consequence of free independence: free convolution 359
5.3.4 Free central limit theorem 368
5.3.5 Freeness for unbounded variables 369
5.4 Link with random matrices 374
5.5 Convergence of the operator norm of polynomials of inde-
pendent GUE matrices 394
5.6 Bibliographical notes 410
Appendices 414
A Linear algebra preliminaries 414
A.I Identities and bounds 414
A.2 Perturbations for normal and Hermitian matrices 415
A.3 Noncommutative matrix//-norms 416
A.4 Brief review of resultants and discriminants 417
B Topological preliminaries 418
B.I Generalities 418
B.2 Topological vector spaces and weak topologies 420
B.3 Banach and Polish spaces 422
B.4 Some elements of analysis 423
C Probability measures on Polish spaces 423
C.I Generalities 423
C.2 Weak topology 425
D Basic notions of large deviations 427
E The skew field H of quaternions and matrix theory over F 430
E. 1 Matrix terminology over F and factorization theorems 431
xii Contents
E.2 The spectral theorem and key corollaries 433
E.3 A specialized result on projectors 434
E.4 Algebra for curvature computations 435
F Manifolds 437
F.I Manifolds embedded in Euclidean space 438
F.2 Proof of the coarea formula 442
F.3 Metrics, connections, curvature, Hessians, and the
Laplace-Beltrami operator 445
G Appendix on operator algebras 450
G. 1 Basic definitions 450
G.2 Spectral properties 452
G.3 States and positivity 454
G.4 von Neumann algebras 455
G.5 Noncommutative functional calculus 457
H Stochastic calculus notions 459
References 465
General conventions and notation 481
Index 484
|
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| spelling | Anderson, Greg W. (DE-588)138987599 aut An introduction to random matrices Greg W. Anderson ; Alice Guionnet ; Ofer Zeitouni first published Cambridge ; New York ; Melbourne ; Madrid ; Cape Town ; SIngapore ; Sao Paulo ; Delhi Cambridge University Press 2010 xiv, 492 Seiten txt rdacontent n rdamedia nc rdacarrier Cambridge studies in advanced mathematics 118 Random matrices Stochastische Matrix (DE-588)4057624-3 gnd rswk-swf Stochastische Matrix (DE-588)4057624-3 s DE-604 Guionnet, Alice (DE-588)137799268 aut Zaitûnî, ʿOfer (DE-588)120906058 aut Cambridge studies in advanced mathematics 118 (DE-604)BV000003678 118 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017739352&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Anderson, Greg W. Guionnet, Alice Zaitûnî, ʿOfer An introduction to random matrices Cambridge studies in advanced mathematics Random matrices Stochastische Matrix (DE-588)4057624-3 gnd |
| subject_GND | (DE-588)4057624-3 |
| title | An introduction to random matrices |
| title_auth | An introduction to random matrices |
| title_exact_search | An introduction to random matrices |
| title_full | An introduction to random matrices Greg W. Anderson ; Alice Guionnet ; Ofer Zeitouni |
| title_fullStr | An introduction to random matrices Greg W. Anderson ; Alice Guionnet ; Ofer Zeitouni |
| title_full_unstemmed | An introduction to random matrices Greg W. Anderson ; Alice Guionnet ; Ofer Zeitouni |
| title_short | An introduction to random matrices |
| title_sort | an introduction to random matrices |
| topic | Random matrices Stochastische Matrix (DE-588)4057624-3 gnd |
| topic_facet | Random matrices Stochastische Matrix |
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