Large random matrices: lectures on macroscopic asymptotics ; École d'Été de Probabilités de Saint-Flour XXXVI - 2006
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| Main Author: | |
|---|---|
| Format: | Conference Proceeding Book |
| Language: | English |
| Published: |
Berlin [u.a.]
Springer
2009
|
| Series: | Lecture notes in mathematics
1957 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Literaturverz. S. 275 - 285 |
| Physical Description: | XII, 294 S. graph. Darst. |
| ISBN: | 9783540698968 |
Staff View
MARC
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| 245 | 1 | 0 | |a Large random matrices |b lectures on macroscopic asymptotics ; École d'Été de Probabilités de Saint-Flour XXXVI - 2006 |c Alice Guionnet |
| 264 | 1 | |a Berlin [u.a.] |b Springer |c 2009 | |
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Record in the Search Index
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|---|---|
| adam_text | Contents
Introduction
................................................... 1
Part
I
Wigner Matrices
and Moments Estimates
5
1
Wigner s Theorem
......................................... 7
1.1
Catalan
Numbers, Non-crossing
Partitions
and Dick Paths
.... 7
1.2
Wigner s
Theorem.......................................
16
1.3
Weak
Convergence
of the Spectral Measure
................. 20
1.4
Relaxation of the Hypotheses over the Entries-Universality
... 22
2
Wigner s Matrices; More Moments Estimates
.............. 29
2.1
Central Limit Theorem
................................... 29
2.2
Estimates of the Largest Eigenvalue of Wigner Matrices
...... 33
3
Words in Several Independent Wigner Matrices
........... 41
3.1
Partitions of Colored Elements and Stars
................... 41
3.2
Voiculescu s Theorem
.................................... 42
Part II Wigner Matrices and Concentration Inequalities
47
4
Concentration Inequalities and Logarithmic Sobolev
Inequalities
................................................ 49
4.1
Concentration Inequalities for Laws Satisfying Logarithmic
Sobolev Inequalities
...................................... 49
4.2
A Few Laws Satisfying a Log-Sobolev Inequality
............. 52
VII
VIII Contents
5
Generalizations
............................................ 59
5.1
Concentration Inequalities for Laws Satisfying Weaker
Coercive Inequalities
..................................... 59
5.2
Concentration Inequalities by Talagrand s Method
........... 60
5.3
Concentration Inequalities on Compact Riemannian Manifold
with Positive
Ricci
Curvature
............................. 61
5.4
Local Concentration Inequalities
........................... 62
6
Concentration Inequalities for Random Matrices
........... 65
6.1
Smoothness and Convexity of the Eigenvalues
of a Matrix
............................................. 65
6.2
Concentration Inequalities for the Eigenvalues
of Random Matrices
..................................... 70
6.3
Concentration Inequalities for Traces of Several Random
Matrices
............................................... 72
6.4
Concentration Inequalities for the
Haar
Measure
on 0{N)
............................................... 74
6.5
Brascamp-Lieb Inequalities; Applications
to Random Matrices
.................................... 77
Part III Matrix Models
89
7
Maps and Gaussian Calculus
.............................. 93
7.1
Combinatorics of Maps and Non-commutative
Polynomials
............................................ 93
7.2
Non-commutative Polynomials
............................ 93
7.3
Maps and Polynomials
................................... 97
7.4
Formal Expansion of Matrix Integrals
...................... 99
8
First-order Expansion
.....................................109
8.1
Finite-dimensional Schwinger-Dyson Equations
.............109
8.2
Tightness and Limiting Schwinger-Dyson Equations
.........110
8.3
Convergence of the Empirical Distribution
..................
H3
8.4
Combinatorial Interpretation of the Limit
..................114
8.5
Convergence of the Free Energy
...........................118
9
Second-order Expansion for the Free Energy
...............121
9.1
Rough Estimates on the Size of the Correction
б?
...........122
9.2
Central Limit Theorem
...................................124
9.3
Comments on the Results
................................137
9.4
Second-order Correction to the Free Energy
................14°
Contents
IX
Part IV Eigenvalues of Gaussian Wigner Matrices
and Large Deviations
147
10
Large Deviations for the Law of the Spectral Measure
of Gaussian Wigner s Matrices
.............................149
11
Large Deviations of the Maximum Eigenvalue
.............159
Part V Stochastic Calculus
165
12
Stochastic Analysis for Random Matrices
.................167
12.1
Dyson s Brownian Motion
................................167
12.2
Itô s
Calculus
...........................................175
12.3
A Dynamical Proof of Wigner s Theorem
1.13...............176
13
Large Deviation Principle for the Law of the Spectral
Measure of Shifted Wigner Matrices
.......................183
13.1
Large Deviations from the Hydrodynamical Limit
for a System of Independent Brownian Particles
.............186
13.2
Large Deviations for the Law of the Spectral Measure
of a Non-centered Large Dimensional Matrix-valued
Brownian Motion
........................................192
14
Asymptotics of Harish—Chandra—It
zykson—Zuber
Integrals and of
Schur
Polynomials
........................211
15
Asymptotics of Some Matrix Integrals
....................217
15.1
Enumeration of Maps from Matrix Models
..................220
15.2
Enumeration of Colored Maps from Matrix Models
..........222
Part VI Free Probability
225
16
Free Probability Setting
..................................227
16.1
A Few Notions about Algebras and Tracial States
...........227
16.2
Space of Laws of
m
Non-commutative Self-adjoint Variables
.. 228
17
Freeness
...................................................231
17.1
Definition of Freeness
....................................231
17.2
Asymptotic Freeness
.....................................232
17.3
The Combinatorics of Freeness
............................236
18
Free Entropy
..............................................245
X
Contents
Part
VII
Appendix
261
19
Basics of Matrices
.........................................263
19.1
Weyľs
and Lidskii s Inequalities
..........................263
19.2
Non-commutative Holder Inequality
......................264
20
Basics of Probability Theory
...............................267
20.1
Basic Notions of Large Deviations
........................267
20.2
Basics of Stochastic Calculus
..............................270
20.3
Proof of
(2.3)...........................................274
References
.....................................................275
Index
..........................................................287
List of Participants of the Summer School
.....................289
List of Short Lectures Given at the Summer School
............293
|
| any_adam_object | 1 |
| author | Guionnet, Alice |
| author_GND | (DE-588)137799268 |
| author_facet | Guionnet, Alice |
| author_role | aut |
| author_sort | Guionnet, Alice |
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| building | Verbundindex |
| bvnumber | BV035440659 |
| classification_rvk | SI 850 |
| classification_tum | MAT 467f MAT 155f MAT 600f MAT 055f |
| ctrlnum | (OCoLC)318643073 (DE-599)DNB992870658 |
| dewey-full | 519.2 512.9434 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 519 - Probabilities and applied mathematics 512 - Algebra |
| dewey-raw | 519.2 512.9434 |
| dewey-search | 519.2 512.9434 |
| dewey-sort | 3519.2 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Conference Proceeding Book |
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| id | DE-604.BV035440659 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T21:35:19Z |
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| isbn | 9783540698968 |
| language | English |
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| series | Lecture notes in mathematics |
| series2 | Lecture notes in mathematics |
| spelling | Guionnet, Alice Verfasser (DE-588)137799268 aut Large random matrices lectures on macroscopic asymptotics ; École d'Été de Probabilités de Saint-Flour XXXVI - 2006 Alice Guionnet Berlin [u.a.] Springer 2009 XII, 294 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Lecture notes in mathematics 1957 Literaturverz. S. 275 - 285 Stochastische Matrix (DE-588)4057624-3 gnd rswk-swf (DE-588)1071861417 Konferenzschrift 2006 Saint-Flour gnd-content Stochastische Matrix (DE-588)4057624-3 s DE-604 Ecole d'Eté de Probabilités 36 2006 Saint-Flour Sonstige (DE-588)6521461-4 oth Lecture notes in mathematics 1957 (DE-604)BV000676446 1957 Digitalisierung TU Muenchen application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017360929&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Guionnet, Alice Large random matrices lectures on macroscopic asymptotics ; École d'Été de Probabilités de Saint-Flour XXXVI - 2006 Lecture notes in mathematics Stochastische Matrix (DE-588)4057624-3 gnd |
| subject_GND | (DE-588)4057624-3 (DE-588)1071861417 |
| title | Large random matrices lectures on macroscopic asymptotics ; École d'Été de Probabilités de Saint-Flour XXXVI - 2006 |
| title_auth | Large random matrices lectures on macroscopic asymptotics ; École d'Été de Probabilités de Saint-Flour XXXVI - 2006 |
| title_exact_search | Large random matrices lectures on macroscopic asymptotics ; École d'Été de Probabilités de Saint-Flour XXXVI - 2006 |
| title_full | Large random matrices lectures on macroscopic asymptotics ; École d'Été de Probabilités de Saint-Flour XXXVI - 2006 Alice Guionnet |
| title_fullStr | Large random matrices lectures on macroscopic asymptotics ; École d'Été de Probabilités de Saint-Flour XXXVI - 2006 Alice Guionnet |
| title_full_unstemmed | Large random matrices lectures on macroscopic asymptotics ; École d'Été de Probabilités de Saint-Flour XXXVI - 2006 Alice Guionnet |
| title_short | Large random matrices |
| title_sort | large random matrices lectures on macroscopic asymptotics ecole d ete de probabilites de saint flour xxxvi 2006 |
| title_sub | lectures on macroscopic asymptotics ; École d'Été de Probabilités de Saint-Flour XXXVI - 2006 |
| topic | Stochastische Matrix (DE-588)4057624-3 gnd |
| topic_facet | Stochastische Matrix Konferenzschrift 2006 Saint-Flour |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017360929&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000676446 |
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