Random Matrices High Dimensional Phenomena

This book focuses on the behaviour of large random matrices. Standard results are covered, and the presentation emphasizes elementary operator theory and differential equations, so as to be accessible to graduate students and other non-experts. The introductory chapters review material on Lie groups and probability measures in a style suitable for applications in random matrix theory. Later chapters use modern convexity theory to establish subtle results about the convergence of eigenvalue distributions as the size of the matrices increases. Random matrices are viewed as geometrical objects with large dimension. The book analyzes the concentration of measure phenomenon, which describes how measures behave on geometrical objects with large dimension. To prove such results for random matrices, the book develops the modern theory of optimal transportation and proves the associated functional inequalities involving entropy and information. These include the logarithmic Sobolev inequality, which measures how fast some physical systems converge to equilibrium. "The book under review is somewhat special in that it is not so much an introduction to the standard models and topics of random matrix theory, but rather to a set of functional analytic issues that are relevant to random matrices." Michael Stolz, Mathematical ReviewsIntroduction; 1. Metric Measure spaces; 2. Lie groups and matrix ensembles; 3. Entropy and concentration of measure; 4. Free entropy and equilibrium; 5. Convergence to equilibrium; 6. Gradient ows and functional inequalities; 7. Young tableaux; 8. Random point fields and random matrices; 9. Integrable operators and differential equations; 10. Fluctuations and the Tracy-Widom distribution; 11. Limit groups and Gaussian measures; 12. Hermite polynomials; 13. From the Ornstein-Uhlenbeck process to Burger's equation; 14. Noncommutative probability spaces; References; Index.