The Random Matrix Theory of the Classical Compact Groups

Provides a comprehensive introduction to the theory of random orthogonal, unitary, and symplectic matrices.

Elizabeth S. Meckes (Author)

9781108419529, Cambridge University Press

Hardback, published 1 August 2019

224 pages, 11 b/w illus.
23.5 x 15.6 x 1.6 cm, 0.44 kg

'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

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.

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.

Subject Areas: Mathematical physics [PHU], Probability & statistics [PBT], Number theory [PBH]