The Semicircle Law, Free Random Variables and Entropy

This text treats free probability theory, which has been extensively developed since the early 1980s. The emphasis is put on entropy and the random matrix model approach. The presentation demonstrates the extensive interrelation between the topics.

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The book treats free probability theory, which has been extensively developed since the early eighties. The emphasis is put on entropy and the random matrix model approach. It is a unique presentation demonstrating the extensive interrelation between the topics. Wigner's theorem and its broad generalizations, such as asymptotic freeness of independent matrices, are explained in detail. Consistent throughout is the parallelism between the normal and semicircle laws. The authors present Voiculescu's multivariate free entropy theory with full proofs and extend the results to unitary operators. Some applications to operator algebras are also given. The book is the first essentially full-scale presentation on free probability theory and includes improvements of results and proofs in current literature. The combinatorial aspects of the specialized topics are emphasized; many examples are given. The book would be a suitable text for graduate courses in free probability theory.