Random Matrices and Non-Commutative Probability (Hardcover)

Classical independence, moments and cumulants Classical independence CLT a cumulants Cumulants to moments Moments to cumulants, the Mobius function Classical Isserlis’ formula Exercises Non-commutative probability 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 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 Convergence Algebraic convergence Free central limit theorem Free Poisson convergence Sums of triangular arrays Exercises Transforms Stieltjes transform R transform Interrelation S-transform Free infinite divisibility Exercises C* -probability space C* -probability space Spectrum Distribution of a self-adjoint element Free product of C* -probability spaces Free additive and multiplicative convolution Exercises Random matrices Empirical spectral measure Limiting spectral measure Moment and trace Some important matrices A unified treatment Exercises Convergence of some important matrices 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 Joint convergence I: single pattern 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 Joint convergence II: multiple patterns Multiple patterns: colors and indices Joint convergence Two or more patterns at a time Sum of independent patterned matrices Discussion Exercises Asymptotic freeness of random matrices 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 Brown measure Exercises Tying three loose ends Mobius function on NC(n) Equivalence of two freeness definitions Free product construction Exercises Bibliography Index