Free probability theory. And its avatars in representation theory, random matrices, and operator algebras; also featuring: non-commutative distributions (Q512222)

The author presents an introduction to free probability theory. He starts with two examples on group representations and eigenvalues of sums of matrices and discusses their asymptotics. He goes on to define noncommutative probability spaces and free independence, the free convolution and the \(\mathcal{R}\)-transform and shows the additivity of the \(\mathcal{R}\)-transform under free convolutions. He also discusses the free central limit theorem, which involves the semicircle distibution rather than the normal distribution. Other topics are the combinatorial aspects of freeness in terms of noncrossing partitions and free cumulants, asymptotic freeness and the link with the eigenvalue distribution of random matrices, another look at group representations (making the observations in the initial example precise), the origin of freeness in the theory of operator algebras (the isomorphism problem for factors associated to free groups) and, finally, an outlook on noncommutative distributions.