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E-raamat: Random Matrix Theory of the Classical Compact Groups

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(Case Western Reserve University, Ohio)
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Random Matrix Theory of the Classical Compact GroupsSuurem pilt
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This book provides a broad overview of foundational results and recent progress in the study of random matrices from the classical compact groups. Designed to present a complete picture to researchers of different fields, the material makes connections to geometry, analysis, algebra, physics, and statistics.

This is the first book to provide a comprehensive overview of foundational results and recent progress in the study of random matrices from the classical compact groups, drawing on the subject's deep connections to geometry, analysis, algebra, physics, and statistics. The book sets a foundation with an introduction to the groups themselves and six different constructions of Haar measure. Classical and recent results are then presented in a digested, accessible form, including the following: results on the joint distributions of the entries; an extensive treatment of eigenvalue distributions, including the Weyl integration formula, moment formulae, and limit theorems and large deviations for the spectral measures; concentration of measure with applications both within random matrix theory and in high dimensional geometry; and results on characteristic polynomials with connections to the Riemann zeta function. This book will be a useful reference for researchers and an accessible introduction for students in related fields.

Arvustused

'This beautiful book describes an important area of mathematics, concerning random matrices associated with the classical compact groups, in a highly accessible and engaging way. It connects a broad range of ideas and techniques, from analysis, probability theory, and representation theory to recent applications in number theory. It is a really excellent introduction to the subject.' J. P. Keating, University of Bristol 'Meckes's new text is a wonderful contribution to the mathematics literature The book addresses many important topics related to the field of random matrices and provides a who's-who list for the subject in its list of references. Those actively researching in this area should acquire a copy of the book; they will understand the jargon from compact matrix groups, measure theory, and probability ' A. Misseldine, Choice ' the author provides an overview of foundational results and recent progress in the study of random matrices from classical compact groups, that is O(n), U(n) and Sp(2n). The main goal is to answer the general question: 'What is a random orthogonal, unitary or symplectic matrix like'?' Andreas Arvanitoyeorgos, zbMATH ' this is a useful book which can serve both as a reference and as a supplemental reading for a course in random matrices.' Vladislav Kargin, Mathematical Reviews Clippings 'The book makes for a wonderful companion to a topics class on random matrices, and an instructor can easily use it either as a stand-alone text or as complementing other textbooks.' Ofer Zeitouni, Bulletin of the American Mathematical Society

Muu info

Provides a comprehensive introduction to the theory of random orthogonal, unitary, and symplectic matrices.
Preface ix
1 Haar Measure on the Classical Compact Matrix Groups
1(30)
1.1 The Classical Compact Matrix Groups
1(6)
1.2 Haar Measure
7(10)
1.3 Lie Group Structure and Character Theory
17(14)
2 Distribution of the Entries
31(29)
2.1 Introduction
31(7)
2.2 The Density of a Principal Submatrix
38(4)
2.3 How Much Is a Haar Matrix Like a Gaussian Matrix?
42(11)
2.4 Arbitrary Projections
53(7)
3 Eigenvalue Distributions: Exact Formulas
60(30)
3.1 The Weyl Integration Formula
60(11)
3.2 Determinantal Point Processes
71(9)
3.3 Matrix Moments
80(5)
3.4 Patterns in Eigenvalues: Powers of Random Matrices
85(5)
4 Eigenvalue Distributions: Asymptotics
90(41)
4.1 The Eigenvalue Counting Function
90(16)
4.2 The Empirical Spectral Measure and Linear Eigenvalue Statistics
106(7)
4.3 Patterns in Eigenvalues: Self-Similarity
113(4)
4.4 Large Deviations for the Empirical Spectral Measure
117(14)
5 Concentration of Measure
131(30)
5.1 The Concentration of Measure Phenomenon
131(2)
5.2 Logarithmic Sobolev Inequalities and Concentration
133(8)
5.3 The Bakry-Emery Criterion and Concentration for the Classical Compact Groups
141(12)
5.4 Concentration of the Spectral Measure
153(8)
6 Geometric Applications of Measure Concentration
161(20)
6.1 The Johnson-Lindenstrauss Lemma
161(4)
6.2 Dvoretzky's Theorem
165(5)
6.3 A Measure-Theoretic Dvoretzky Theorem
170(11)
7 Characteristic Polynomials and Connections to the Riemann C-function
181(24)
7.1 Two-Point Correlations and Montgomery's Conjecture
181(5)
7.2 The Zeta Function and Characteristic Polynomials of Random Unitary Matrices
186(11)
7.3 Numerical and Statistical Work
197(8)
References 205(7)
Index 212
Elizabeth S. Meckes is Professor of Mathematics at Case Western Reserve University, Ohio. She is a mathematical probabilist specializing in random matrix theory and its applications to other areas of mathematics, physics and statistics. She received her Ph.D. at Stanford University in 2006 and received the American Institute of Mathematics five-year fellowship. She has also received funding from the Clay Institute of Mathematics, the Simons Foundation, and the US National Science Foundation. She is the author of twenty-two research papers in mathematics, as well as the textbook Linear Algebra (Cambridge, 2018), co-authored with Mark Meckes.
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