Random Matrices and Non-Commutative Probability
Chapman and Hall/CRC
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 chapte
Bose, Arup, "Random Matrices and Non-Commutative Probability" (2021). ISI Best Publications. 28.