Lectures on the Combinatorics of Free Probability

This 2006 book is a self-contained introduction to free probability theory suitable for an introductory graduate level course.

Alexandru Nica (Author), Roland Speicher (Author)

9780521858526, Cambridge University Press

Paperback, published 7 September 2006

434 pages, 124 exercises
22.9 x 15.2 x 2.5 cm, 0.63 kg

"This book is an absolutely indispensable resource for anyone who works in free probability theory, as well as those (advanced undergraduates or graduate students, or professional researchers) newly initiated in the field. Lectures on the Combinatorics of Free Probabilityis sure to be considered one of the important informative texts in this rapidly growing field of study, and it is highly recommended for any reader or researcher in the field."
Todd Kemp, Mathematical Reviews

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 2006 book gives 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.

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: Möbius 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.

Subject Areas: Combinatorics & graph theory [PBV], Calculus & mathematical analysis [PBK]