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.
General
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