Random Matrices and Non-Commutative 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.