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E-grāmata: Large Random Matrices: Lectures on Macroscopic Asymptotics: Ecole d'Ete de Probabilites de Saint-Flour XXXVI - 2006

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Large Random Matrices: Lectures on Macroscopic Asymptotics: Ecole d'Ete de Probabilites de Saint-Flour XXXVI - 2006Palielināt
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Random matrix theory has developed in the last few years, in connection with various fields of mathematics and physics. These notes emphasize the relation with the problem of enumerating complicated graphs, and the related large deviations questions. Such questions are also closely related with the asymptotic distribution of matrices, which is naturally defined in the context of free probability and operator algebra. The material of this volume is based on a series of nine lectures given at the Saint-Flour Probability Summer School 2006. Lectures were also given by Maury Bramson and Steffen Lauritzen.

Recenzijas

From the reviews:

This book is a set of lecture notes on eigenvalues of large random matrices. useful to all mathematicians and statisticians who are interested in Wigner matrices. In summary, the book is very much worth perusal. (Vladislav Kargin, Mathematical Reviews, Issue 2010 d)

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
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 O(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
113
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 δNt
122
9.2 Central Limit Theorem
124
9.3 Comments on the Results
137
9.4 Second-order Correction to the Free Energy
140
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 Ito'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–Itzykson–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 in 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
Part VII Appendix 261
19 Basics of Matrices
263
19.1 Weyl's 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
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