Random Matrices and Non-Commutative Probability

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.