The semicircle law, free random variables and entropy / Fumio Hiai, Dénes Petz
Type de document : MonographieCollection : Mathematical surveys and monographs, 77Langue : anglais.Pays: Etats Unis.Éditeur : Providence (R.I.) : American Mathematical Society, 2000Description : 1 vol. (X-376 p.) : fig. ; 26 cmISBN: 0821820818.ISSN: 0885-4653.Bibliographie : Bibliogr. p. 357-369. Index.Sujet MSC : 46L54, Functional analysis - Selfadjoint operator algebras, Free probability and free operator algebras60F10, Limit theorems in probability theory, Large deviations
94A17, Communication, information, Measures of information, entropy
46N50, Miscellaneous applications of functional analysis, Applications in quantum physics
60J65, Probability theory and stochastic processes - Markov processes, Brownian motionEn-ligne : Zentralblatt | MathSciNet
| Item type | Current library | Call number | Status | Date due | Barcode |
|---|---|---|---|---|---|
Monographie
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CMI Salle 1 | 46 HIA (Browse shelf(Opens below)) | Available | 12343-01 |
Bibliogr. p. 357-369. Index
From the authors’ abstract: This is an expository monograph on free probability theory. The emphasis is put on entropy and random matrix models. The highlight is the very far reaching interrelation of free probability and random matrix theories. Wigner’s theorem and its broad generalizations, such as asymptotic freeness of independent matrices are expounded in detail. The parallelism between the normal and semicircle laws runs through the book. Many examples are included to illustrate the results. The frequent random matrix ensembles are characterized by maximization of their Boltzmann-Gibbs entropy under certain constraints, and the asymptotic eigenvalue distribution is treated in the almost everywhere sense and in the form of large deviations. Voiculescu’s multivariate free entropy is presented with full proofs and extended to unitary operators. Some ideas about applications to operator algebras are also given.
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