Random matrices: high dimensional phenomena
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Cambridge [u.a.]
Cambridge Univ. Press
2009
|
| Ausgabe: | 1. publ. |
| Schriftenreihe: | London Mathematical Society lecture note series
367 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis Klappentext |
| Beschreibung: | X, 437 S. |
| ISBN: | 9780521133128 |
Internformat
MARC
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| 245 | 1 | 0 | |a Random matrices |b high dimensional phenomena |c Gordon Blower |
| 250 | |a 1. publ. | ||
| 264 | 1 | |a Cambridge [u.a.] |b Cambridge Univ. Press |c 2009 | |
| 300 | |a X, 437 S. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a London Mathematical Society lecture note series |v 367 | |
| 650 | 4 | |a Random matrices | |
| 650 | 0 | 7 | |a Stochastische Matrix |0 (DE-588)4057624-3 |2 gnd |9 rswk-swf |
| 689 | 0 | 0 | |a Stochastische Matrix |0 (DE-588)4057624-3 |D s |
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Datensatz im Suchindex
| _version_ | 1804140740434984960 |
|---|---|
| adam_text | Contents
Introduction page-
1
Metric
measure spaces
4
1.1
Weak convergence on compact metric spaces
4
1.2
Invariant measure on a compact metric group
10
1.3
Measures on non-compact Polish spaces
16
1.4
The
Brunn -Minkowski
inequality
22
1.5
Gaussian measures
25
1.6
Surface area measure on the spheres
27
1.7
Lipschitz functions and the Hausdorff metric
31
1.8
Characteristic functions and Cauchy transforms
33
Lie groups and matrix ensembles
42
2.1
The classical groups, their eigenvalues and norms
42
2.2
Determinants and functional calculus
49
2.3
Linear Lie groups
56
2.4
Connections and curvature
63
2.5
Generalized ensembles
66
2.6
The Weyl integration formula
72
2.7
Dyson s circular ensembles
78
2.8
Circular orthogonal ensemble
81
2.9
Circular symplectic ensemble
83
Entropy and concentration of measure
84
3.1
Relative entropy
84
3.2
Concentration of measure
93
3.3
Transportation
99
3.4
Transportation inequalities
103
3.5
Transportation inequalities for uniformly
convex potentials
106
3.6
Concentration of measure in matrix ensembles
109
vn
viii Contents
3.7
Concentration
for rectangular Gaussian matrices
114
3.8
Concentration on the sphere
123
3.9
Concentration for compact Lie groups
126
4
Free entropy and equilibrium
132
4.1
Logarithmic energy and equilibrium measure
132
4.2
Energy spaces on the disc
134
4.3
Free versus classical entropy on the spheres
142
4.4
Equilibrium measures for potentials on the real line
147
4.5
Equilibrium densities for convex potentials
154
4.6
The quartic model with positive leading term
159
4.7
Quartic models with negative leading term
164
4.8
Displacement convexity and relative free entropy
169
4.9
Toeplitz determinants
172
5
Convergence to equilibrium
177
5.1
Convergence to arclength
177
5.2
Convergence of ensembles
179
5.3
Mean field convergence
183
5.4
Almost sure weak convergence for uniformly
convex potentials
189
5.5
Convergence for the singular numbers from the
Wishart
distribution
193
6
Gradient flows and functional inequalities
196
6.1
Variation of functionals and gradient flows
196
6.2
Logarithmic Sobolev inequalities
203
6.3
Logarithmic Sobolev inequalities for uniformly
convex potentials
206
6.4
Fisher s information and Shannon s entropy
210
6.5
Free information and entropy
213
6.6
Free logarithmic Sobolev inequality
218
6.7
Logarithmic Sobolev and spectral gap inequalities
221
6.8
inequalities for Gibbs measures on
Riemannian manifolds
223
7
Young tableaux
227
7.1
Group representations
227
7.2
Young diagrams
229
7.3
The Vershik
Ω
distribution
237
7.4
Distribution of the longest increasing subsequence
243
7.5
Inclusion-exclusion principle
250
Contents ix
8
Random point fields and random matrices
253
8.1
Determinantal random point fields
253
8.2
Determinantal random point fields on the real line
261
8.3
Determinantal random point fields and orthogonal
polynomials
270
8.4 De Branges s
spaces
274
8.5
Limits of kernels
278
9
Integrable
operators and differential equations
281
9.1
Integrable
operators and Hankel integral operators
281
9.2
Hankel integral operators that commute with second
order differential operators
289
9.3
Spectral bulk and the nine kernel
293
9.4
Soft edges and the Airy kernel
299
9.5
Hard edges and the Bessel kernel
304
9.6
The spectra of Hankel operators and rational
approximation
310
9.7
The Tracy
Widom
distribution
315
10
Fluctuations and the Tracy-Widom distribution
321
10.1
The
Costili Lebowitz
central limit theorem
321
10.2
Discrete Tracy
Widom
systems
327
10.3
The discrete Bessel kernel
328
10.4
Plancherel measure on the partitions
334
10.5
Fluctuations of the longest increasing subsequence
343
10.6
Fluctuations of linear statistics over unitary
ensembles
345
11
Limit groups and Gaussian measures
352
11.1
Some inductive limit groups
352
11.2
Hua
Pickrell measure on the infinite unitary group
357
11.3
Gaussian Hubert space
365
11.4
Gaussian measures and fluctuations
369
12
Hermite polynomials
373
12.1
Tensor products of Hubert space
373
12.2
Hermite polynomials and Mehler s formula
375
12.3
The Ornstein-Uhlenbeck semigroup
381
12.4
Hermite polynomials in higher dimensions
384
13
From the Ornstein-Uhlenbeck process to the
Burgers equation
392
13.1
The Ornstein Uhlenbeck process
392
χ
Contents
13.2
The logarithmic Sobolev inequality for the
Ornstein-Uhlenbeck generator
396
13.3
The matrix Ornstein-Uhlenbeck process
398
13.4
Solutions for matrix stochastic differential equations
401
13.5
The Burgers equation
408
14
Noncommutative probability spaces
411
14.1
Noncommutative probability spaces
411
14.2
Tracial probability spaces
414
14.3
The semicircular distribution
418
References
424
Index
433
LONDON
MATHEMATICAL SOCIETY
LECTURE NOTE SERIES
Edited by Professor M. Rf
i d
Mathematics
ЈпѕШте,
iveatry
CV4
7
AL
United Kingdom
with the assistance of
B. J. Green (Cambridge)
D. R.
Heath-Brown (Oxford)
R. A. M. Rouquier (Oxford)
J. T. Stafford {Manchester)
E.
Süli
(Oxford)
The London Mathematical Society
ix
incorporated under Royal Charter.
Random Matrices: High Dimensional Phenomena
Gordon Blower
This book focuses on the behaviour of large random matrices. Standard results
are covered, and the presentation emphasizes elementary operator theory and
differential equations, so as to be accessible to graduate students and other
non-experts. The introductory chapters review material on Lie groups and
probability measures in a style suitable for applications in random matrix
theory. Later chapters use modern convexity theory to establish subtle results
about the convergence of eigenvalue distributions as the size of the matrices
increases.
Random matrices are viewed as geometrical objects with large dimension.
The book analyses the concentration of measure phenomenon, which describes
how measures behave on geometrical objects with large dimension. To prove
such results for random matrices, the book develops the modern theory of
optimal transportation and proves the associated functional inequalities
involving entropy and information. These include the logarithmic Sobolev
inequality, which measures how fast some physical systems converge to
equilibrium.
Cambridge
UNIVERSITY PRESS
www.cambridge.org
ISBN
978-0-521-13312-8
9 780521 133128 >
Fora Ibi
ofbooks available in this series, seepage
L
|
| any_adam_object | 1 |
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| discipline | Mathematik |
| edition | 1. publ. |
| format | Book |
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| spelling | Blower, G. 1966- Verfasser (DE-588)1214248322 aut Random matrices high dimensional phenomena Gordon Blower 1. publ. Cambridge [u.a.] Cambridge Univ. Press 2009 X, 437 S. txt rdacontent n rdamedia nc rdacarrier London Mathematical Society lecture note series 367 Random matrices Stochastische Matrix (DE-588)4057624-3 gnd rswk-swf Stochastische Matrix (DE-588)4057624-3 s DE-604 London Mathematical Society lecture note series 367 (DE-604)BV000000130 367 Digitalisierung UB Bayreuth application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=018655813&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis Digitalisierung UB Bayreuth application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=018655813&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Klappentext |
| spellingShingle | Blower, G. 1966- Random matrices high dimensional phenomena London Mathematical Society lecture note series Random matrices Stochastische Matrix (DE-588)4057624-3 gnd |
| subject_GND | (DE-588)4057624-3 |
| title | Random matrices high dimensional phenomena |
| title_auth | Random matrices high dimensional phenomena |
| title_exact_search | Random matrices high dimensional phenomena |
| title_full | Random matrices high dimensional phenomena Gordon Blower |
| title_fullStr | Random matrices high dimensional phenomena Gordon Blower |
| title_full_unstemmed | Random matrices high dimensional phenomena Gordon Blower |
| title_short | Random matrices |
| title_sort | random matrices high dimensional phenomena |
| title_sub | high dimensional phenomena |
| topic | Random matrices Stochastische Matrix (DE-588)4057624-3 gnd |
| topic_facet | Random matrices Stochastische Matrix |
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