Description
This is an introductory book on Non-Commutative Probability or Free Probability and Large Dimensional Random Matrices. Basic concepts of free probability are introduced by analogy with classical probability in a lucid and quick manner. It then develops the results on the convergence of large dimensional random matrices, with a special focus on the interesting connections to free probability. The book assumes almost no prerequisite for the most part. However, familiarity with the basic convergence concepts in probability and a bit of mathematical maturity will be helpful.
- Combinatorial properties of non-crossing partitions, including the Möbius function play a central role in introducing free probability.
- Free independence is defined via free cumulants in analogy with the way classical independence can be defined via classical cumulants.
- Free cumulants are introduced through the Möbius function.
- Free product probability spaces are constructed using free cumulants.
- Marginal and joint tracial convergence of large dimensional random matrices such as the Wigner, elliptic, sample covariance, cross-covariance, Toeplitz, Circulant and Hankel are discussed.
- Convergence of the empirical spectral distribution is discussed for symmetric matrices.
- Asymptotic freeness results for random matrices, including some recent ones, are discussed in detail. These clarify the structure of the limits for joint convergence of random matrices.
- Asymptotic freeness of independent sample covariance matrices is also demonstrated via embedding into Wigner matrices.
- Exercises, at advanced undergraduate and graduate level, are provided in each chapter.
Table of Contents
- Classical independence, moments and cumulants
- Non-commutative probability
- Free independence
- Convergence
- Transforms
- C* -probability space
- Random matrices
- Convergence of some important matrices
- Joint convergence I: single pattern
- Joint convergence II: multiple patterns
- Asymptotic freeness of random matrices
- Brown measure
- Tying three loose ends
Classical independence
CLT a cumulants
Cumulants to moments
Moments to cumulants, the Möbius function
Classical Isserlis’ formula
Exercises
Non-crossing partition
Free cumulants
Free Gaussian or semi-circle law
Free Poisson law
Non-commutative and *-probability spaces
Moments and probability laws of variables
Exercises
Free independence
Free product of *-probability spaces
Free binomial
Semi-circular family
Free Isserlis’ formula
Circular and elliptic variables
Free additive convolution
Kreweras complement
Moments of free variables
0 Compound free Poisson
Exercises
Algebraic convergence
Free central limit theorem
Free Poisson convergence
Sums of triangular arrays
Exercises
Stieltjes transform
R transform
Interrelation
S-transform
Free infinite divisibility
Exercises
C* -probability space
Spectrum
Distribution of a self-adjoint element
Free product of C* -probability spaces
Free additive and multiplicative convolution
Exercises
Empirical spectral measure
Limiting spectral measure
Moment and trace
Some important matrices
A unified treatment
Exercises
Wigner matrix: semi-circle law
S-matrix: Marcenko-Pastur law
IID and elliptic matrices: circular and elliptic variables
Toeplitz matrix
Hankel matrix
Reverse Circulant matrix: symmetrized Rayleigh law
Symmetric Circulant: Gaussian law
Almost sure convergence of the ESD
Exercises
Unified treatment: extension
Wigner matrices: asymptotic freeness
Elliptic matrices: asymptotic freeness
S-matrices in elliptic models: asymptotic freeness
Symmetric Circulants: asymptotic independence
Reverse Circulants: asymptotic half independence
Exercises
Multiple patterns: colors and indices
Joint convergence
Two or more patterns at a time
Sum of independent patterned matrices
Discussion
Exercises
Elliptic, IID, Wigner, and S-matrices
Gaussian elliptic, IID, Wigner and deterministic matrices
General elliptic, IID, Wigner, and deterministic matrices
S-matrices and embedding
Cross covariance matrices
Pair-correlated cross-covariance; p/n ! y
Pair correlated cross covariance; p/n !
Wigner and patterned random matrices
Discussion
Exercises
Brown measure
Exercises
Möbius function on NC(n)
Equivalence of two freeness definitions
Free product construction
Exercises
Bibliography
Index
