#### Title

Random Matrices and Non-Commutative Probability

#### Document Type

Book

#### Publication Title

Chapman and Hall/CRC

#### Abstract

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

#### First Page

1

#### Last Page

286

#### DOI

10.1201/9781003144496

#### Publication Date

10-27-2021

#### Recommended Citation

Bose, Arup, "Random Matrices and Non-Commutative Probability" (2021). *ISI Best Publications*. 28.

https://digitalcommons.isical.ac.in/ibp/28