The Semicircle Law, Free Random Variables and Entropy

The Semicircle Law, Free Random Variables and Entropy

by Fumio Hiai, Denes Petz
     
 

ISBN-10: 0821841351

ISBN-13: 9780821841358

Pub. Date: 03/23/2006

Publisher: American Mathematical Society

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

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Overview

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.

Product Details

ISBN-13:
9780821841358
Publisher:
American Mathematical Society
Publication date:
03/23/2006
Series:
Mathematical Surveys and Monographs, #77
Pages:
376
Product dimensions:
7.00(w) x 10.00(h) x 0.90(d)

Table of Contents

Prefaceix
Overview1
0.1The isomorphism problem of free group factors1
0.2From the relation of free generators to free probability3
0.3Random matrices5
0.4Entropy and large deviations9
0.5Voiculescu's free entropy for multivariables12
0.6Operator algebras14
1Probability Laws and Noncommutative Random Variables19
1.1Distribution measure of normal operators20
1.2Noncommutative random variables32
2The Free Relation39
2.1The free product of noncommutative probability spaces40
2.2The free relation42
2.3The free central limit theorem48
2.4Free convolution of measures52
2.5Moments and cumulants60
2.6Multivariables71
3Analytic Function Theory and Infinitely Divisible Laws91
3.1Cauchy transform, Poisson integral, and Hilbert transform92
3.2Relation between Cauchy transform and R-series95
3.3Infinitely divisible laws98
4Random Matrices and Asymptotically Free Relation113
4.1Random matrices and their eigenvalues114
4.2Random unitary matrices and asymptotic freeness135
4.3Asymptotic freeness of some random matrices146
4.4Random matrix models of noncommutative random variables161
5Large Deviations for Random Matrices175
5.1Boltzmann entropy and large deviations176
5.2Entropy and random matrices181
5.3Logarithmic energy and free entropy189
5.4Gaussian and unitary random matrices209
5.5The Wishart matrix226
5.6Entropy and large deviations revisited239
6Free Entropy of Noncommutative Random Variables245
6.1Definition and basic properties246
6.2Calculus for power series of noncommutative variables253
6.3Change of variable formulas for free entropy259
6.4Additivity of free entropy269
6.5Free entropies of unitary and non-selfadjoint random variables275
6.6Relation between different free entropies280
7Relation to Operator Algebras301
7.1Free group factors and semicircular systems302
7.2Interpolated free group factors310
7.3Free entropy dimension327
7.4Applications of free entropy346
Bibliography357
Index371

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