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Lectures on the Combinatorics of Free Probability

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ISBN-10: 0521858526

ISBN-13: 9780521858526

Edition: 2006

Authors: Alexandru Nica, Roland Speicher, N. J. Hitchin

List price: $140.00
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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 book is the first to give a self-contained and comprehensive introduction to free probability…    
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Book details

List price: $140.00
Copyright year: 2006
Publisher: Cambridge University Press
Publication date: 9/7/2006
Binding: Paperback
Pages: 434
Size: 6.00" wide x 8.75" long x 1.00" tall
Weight: 1.342
Language: English

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

Basic concepts
Non-commutative probability spaces and distributions
Non-commutative probability spaces
*-distributions (case of normal elements)
*-distributions (general case)
A case study of non-normal distribution
Description of the example
Dyck paths
The distribution of a + a[superscript *]
Using the Cauchy transform
C[superscript *]-probability spaces
Functional calculus in a C[superscript *]-algebra
C[superscript *]-probability spaces
*-distribution, norm and spectrum for a normal element
Non-commutative joint distributions
Joint distributions
Joint *-distributions
Joint *-distributions and isomorphism
Definition and basic properties of free independence
The classical situation: tensor independence
Definition of free independence
The example of a free product of groups
Free independence and joint moments
Some basic properties of free independence
Are there other universal product constructions?
Free product of *-probability spaces
Free product of unital algebras
Free product of non-commutative probability spaces
Free product of *-probability spaces
Free product of C[superscript *]-probability spaces
The GNS representation
Free product of C[superscript *]-probability spaces
Example: semicircular systems and the full Fock space
Motivation: free central limit theorem
Convergence in distribution
General central limit theorem
Classical central limit theorem
Free central limit theorem
The multi-dimensional case
Conclusion and outlook
Basic combinatorics I: non-crossing partitions
Non-crossing partitions of an ordered set
The lattice structure of NC(n)
The factorization of intervals in NC
Basic combinatorics II: Mobius inversion
Convolution in the framework of a poset
Mobius inversion in a lattice
The Mobius function of NC
Multiplicative functions on NC
Functional equation for convolution with [Mu subscript n]
Free cumulants: definition and basic properties
Multiplicative functionals on NC
Definition of free cumulants
Products as arguments
Free independence and free cumulants
Cumulants of random variables
Example: semicircular and circular elements
Even elements
Classical cumulants
Sums of free random variables
Free convolution
Analytic calculation of free convolution
Proof of the free central limit theorem via R-transform
Free Poisson distribution
Compound free Poisson distribution
More about limit theorems and infinitely divisible distributions
Limit theorem for triangular arrays
Cumulants of operators on Fock space
Infinitely divisible distributions
Conditionally positive definite sequences
Characterization of infinitely divisible distributions
Products of free random variables
Multiplicative free convolution
Combinatorial description of free multiplication
Compression by a free projection
Convolution semigroups ([Mu superscript Characters not reproduciblet])[subscript tge]1
Compression by a free family of matrix units
R-diagonal elements
Motivation: cumulants of Haar unitary elements
Definition of R-diagonal elements
Special realizations of tracial R-diagonal elements
Product of two free even elements
The free anti-commutator of even elements
Powers of R-diagonal elements
Transforms and models
The R-transform
The multi-variable R-transform
The functional equation for the R-transform
More about the one-dimensional case
The operation of boxed convolution
The definition of boxed convolution, and its motivation
Basic properties of boxed convolution
Radial series
The Mobius series and its use
More on the one-dimensional boxed convolution
Relation to multiplicative functions on NC
The S-transform
The free commutator
Free commutators of even elements
Free commutators in the general case
The cancelation phenomenon
R-cyclic matrices
Definition and examples of R-cyclic matrices
The convolution formula for an R-cyclic matrix
R-cyclic families of matrices
Applications of the convolution formula
The full Fock space model for the R-transform
Description of the Fock space model
An application: revisiting free compressions
Gaussian random matrices
Moments of Gaussian random variables
Random matrices in general
Selfadjoint Gaussian random matrices and genus expansion
Asymptotic free independence for several independent Gaussian random matrices
Asymptotic free independence between Gaussian random matrices and constant matrices
Unitary random matrices
Haar unitary random matrices
The length function on permutations
Asymptotic freeness for Haar unitary random matrices
Asymptotic freeness between randomly rotated constant matrices
Embedding of non-crossing partitions into permutations
Notes and comments