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Lectures on the Combinatorics of Free Probability

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Lectures on the Combinatorics of Free Probability

Lectures on the Combinatorics of Free Probability

Alexandru Nica, University of Waterloo, Ontario
Roland Speicher, Queen's University, Ontario
September 2006
Available
Paperback
9780521858526
£116.00
GBP
Paperback
USD
eBook
    Description

    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.

    • Presents the state of the art of the combinatorial facet of free probability
    • Gives a friendly and self-contained introduction to the general field of free probability
    • Written in a style which makes it ideal for use in the presentation of a graduate level course

    Product details

    September 2006
    Paperback
    9780521858526
    434 pages
    229 × 152 × 25 mm
    0.63kg
    124 exercises
    Available

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    Contents

    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: 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.
      • Show more

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    About the authors
      Authors
    • Alexandru Nica , University of Waterloo, Ontario

      Alexandru Nica is a Professor of Mathematics at the University of Waterloo, Ontario.

    • Roland Speicher , Queen's University, Ontario

      Roland Speicher is a Professor of Mathematics at Queen's University, Kingston.

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