The Random Matrix Theory of the Classical Compact Groups: (Series Number 218 Cambridge Tracts in Mathematics)

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.

Table of Contents:
1. Haar measure on the classical compact matrix groups; 2. Distribution of the entries; 3. Eigenvalue distributions: exact formulas; 4. Eigenvalue distributions: asymptotics; 5. Concentration of measure; 6. Geometric applications of measure concentration; 7. Characteristic polynomials and the zeta function.

About the Author :
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.

Review :
'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