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Lectures on the Combinatorics of Free Probability

Lectures on the Combinatorics of Free Probability

Part of London Mathematical Society Lecture Note Series

  • Date Published: September 2006
  • availability: Available
  • format: Paperback
  • isbn: 9780521858526

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  • 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
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    Product details

    • Date Published: September 2006
    • format: Paperback
    • isbn: 9780521858526
    • length: 434 pages
    • dimensions: 229 x 152 x 25 mm
    • weight: 0.63kg
    • contains: 124 exercises
    • availability: Available
  • 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.

  • 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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