The Semicircle Law, Free Random Variables and Entropy / Edition 1

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The book 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 volume 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 the book is the parallelism between the normal and semicircle laws. Voiculescu's multivariate free entropy theory is presented with full proofs and extends the results to unitary operators. Some applications to operator algebras are also given. Based on lectures given by the authors in Hungary, Japan, and Italy, the book is a good reference for mathematicians interested in free probability theory and can serve as a text for an advanced graduate course.
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Editorial Reviews

Basing their work on the recent discoveries of Dan Voiculescu, the authors of this volume focus on the role of entropy and random matrix models. Chapters address the connection between the concepts of probability theory and linear operators in Hilbert spaces; the fundamental free relation; the basic random matrix models and the limit of their eigenvalue distribution; entropy; Voiculescu's multivariate free entropy, discussed in the setting of operator algebras; and free group factors, with ideas on applications to operator algebras. The authors attempt to make the material as self-contained as possible. Annotation c. Book News, Inc., Portland, OR (
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Product Details

  • ISBN-13: 9780821820810
  • Publisher: American Mathematical Society
  • Publication date: 1/1/2000
  • Series: Surv Series
  • Edition number: 1
  • Pages: 376

Table of Contents

Preface ix
Overview 1
0.1 The isomorphism problem of free group factors 1
0.2 From the relation of free generators to free probability 3
0.3 Random matrices 5
0.4 Entropy and large deviations 9
0.5 Voiculescu's free entropy for multivariables 12
0.6 Operator algebras 14
1 Probability Laws and Noncommutative Random Variables 19
1.1 Distribution measure of normal operators 20
1.2 Noncommutative random variables 32
2 The Free Relation 39
2.1 The free product of noncommutative probability spaces 40
2.2 The free relation 42
2.3 The free central limit theorem 48
2.4 Free convolution of measures 52
2.5 Moments and cumulants 60
2.6 Multivariables 71
3 Analytic Function Theory and Infinitely Divisible Laws 91
3.1 Cauchy transform, Poisson integral, and Hilbert transform 92
3.2 Relation between Cauchy transform and R-series 95
3.3 Infinitely divisible laws 98
4 Random Matrices and Asymptotically Free Relation 113
4.1 Random matrices and their eigenvalues 114
4.2 Random unitary matrices and asymptotic freeness 135
4.3 Asymptotic freeness of some random matrices 146
4.4 Random matrix models of noncommutative random variables 161
5 Large Deviations for Random Matrices 175
5.1 Boltzmann entropy and large deviations 176
5.2 Entropy and random matrices 181
5.3 Logarithmic energy and free entropy 189
5.4 Gaussian and unitary random matrices 209
5.5 The Wishart matrix 226
5.6 Entropy and large deviations revisited 239
6 Free Entropy of Noncommutative Random Variables 245
6.1 Definition and basic properties 246
6.2 Calculus for power series of noncommutative variables 253
6.3 Change of variable formulas for free entropy 259
6.4 Additivity of free entropy 269
6.5 Free entropies of unitary and non-selfadjoint random variables 275
6.6 Relation between different free entropies 280
7 Relation to Operator Algebras 301
7.1 Free group factors and semicircular systems 302
7.2 Interpolated free group factors 310
7.3 Free entropy dimension 327
7.4 Applications of free entropy 346
Bibliography 357
Index 371
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