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.

Large dimensional random matrices

such as the Wigner, elliptic, sample covariance, cross-covariance, Toeplitz, Circulant and Hankel are discussed, including marginal and joint tracial convergence.

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.