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9780521858526

Lectures on the Combinatorics of Free Probability

by
  • ISBN13:

    9780521858526

  • ISBN10:

    0521858526

  • Edition: 1st
  • Format: Paperback
  • Copyright: 2006-09-11
  • Publisher: Cambridge University Press

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Summary

Free Probability Theory studies a special class of 'noncommutative’random variables, which appear in the context of operators on Hilbert spaces and in one of the large random matrices. Since its emergence in the 1980s, free probability has evolved into an established field of mathematics with strong connections to other mathematical areas, such as operator algebras, classical probability theory, random matrices, combinatorics, representation theory of symmetric groups. Free probability also connects to more applied scientific fields, such as wireless communication in electrical engineering. This book is the first to give a self-contained and comprehensive introduction to free probability theory which has its main focus on the combinatorial aspects. The volume is designed so that it can be used as a text for an introductory course (on an advanced undergraduate or beginning graduate level), and is also well-suited for the individual study of free probability.

Table of Contents

Part I. Basic Concepts: 1. Non-commutative probability spaces and distributions
2. A case study of non-normal distribution
3. C*-probability spaces
4. Non-commutative joint distributions
5. Definition and basic properties of free independence
6. Free product of *-probability spaces
7. Free product of C*-probability spaces
Part II. Cumulants: 8. Motivation: free central limit theorem
9. Basic combinatorics I: non-crossing partitions
10. Basic Combinatorics II: Mö
bius inversion
11. Free cumulants: definition and basic properties
12. Sums of free random variables
13. More about limit theorems and infinitely divisible distributions
14. Products of free random variables
15. R-diagonal elements
Part III. Transforms and Models: 16. The R-transform
17. The operation of boxed convolution
18. More on the 1-dimensional boxed convolution
19. The free commutator
20. R-cyclic matrices
21. The full Fock space model for the R-transform
22. Gaussian Random Matrices
23. Unitary Random Matrices
Notes and Comments
Bibliography
Index.

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