An introduction to random matrices:
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Cambridge [u.a.]
Cambridge Univ. Press
2011
|
| Edition: | 1. publ., reprint. with corr. |
| Series: | Cambridge studies in advanced mathematics
118 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XIV, 491 S. |
| ISBN: | 9780521194525 |
Staff View
MARC
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| 245 | 1 | 0 | |a An introduction to random matrices |c Greg W. Anderson ; Alice Guionnet ; Ofer Zeitouni |
| 250 | |a 1. publ., reprint. with corr. | ||
| 264 | 1 | |a Cambridge [u.a.] |b Cambridge Univ. Press |c 2011 | |
| 300 | |a XIV, 491 S. | ||
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Record in the Search Index
| _version_ | 1804148551139196928 |
|---|---|
| adam_text | 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
viii 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
3
Hermite 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 LN
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 at
0
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
β
ensembles
302
4.5.1
Tridiagonal representation of
β
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
U
-norms
416
A.4 Brief review of resultants and discriminants
417
В
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
С
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.
1
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
483
|
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| spelling | Anderson, Greg W. Verfasser (DE-588)138987599 aut An introduction to random matrices Greg W. Anderson ; Alice Guionnet ; Ofer Zeitouni 1. publ., reprint. with corr. Cambridge [u.a.] Cambridge Univ. Press 2011 XIV, 491 S. 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 Verfasser (DE-588)137799268 aut Zaitûnî, ʿOfer Verfasser (DE-588)120906058 aut Cambridge studies in advanced mathematics 118 (DE-604)BV000003678 118 Digitalisierung UB Bayreuth application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=024533794&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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