Random Matrices and Non-Commutative Probability

<p>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. </p><ul> <p> </p> <li>Combinatorial properties of non-crossing partitions including the Möbius function play a central role in introducing free probability. </li> </ul><ul> <p> </p> <li>Free independence is defined via free cumulants in analogy with the way classical independence can be defined via classical cumulants.</li> </ul><ul> <p> </p> <li>Free cumulants are introduced through the Möbius function. </li> </ul><ul> <p> </p> <li>Free product probability spaces are constructed using free cumulants. </li> </ul><ul> <p> </p> <li>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. </li> </ul><ul> <p> </p> <li>Convergence of the empirical spectral distribution is discussed for symmetric matrices. </li> </ul><ul> <p> </p> <li>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. </li> </ul><ul> <p> </p> <li>Asymptotic freeness of independent sample covariance matrices is also demonstrated via embedding into Wigner matrices. </li> </ul><ul> <p> </p> <li>Exercises at advanced undergraduate and graduate level are provided in each chapter. </li> </ul>