Free Probability Theory and Random Matrices

Free probability theory originated in the context of operator algebras, however, one of the main features of that theory is its connection with random matrices. Indeed, free probability can be considered as the theory providing concepts and notations, without relying on random matrices, for dealing with the limit N → ∞ of N ×N -random matrices. One of the basic approaches to free probability, on which I will concentrate in this lecture, is of a combinatorial nature and centers around so-called free cumulants. In the spirit of the above these arise as the combinatorics (in leading order) of N ×N -random matrices in the limit N = ∞. These free cumulants are multi-linear functionals which are defined in combinatorial terms by a formula involving noncrossing partitions. I will present the basic definitions and properties of non-crossing partitions and free cumulants and outline its relations with freeness and random matrices. As examples, I will consider the problems of calculating the eigenvalue distribution of the sum of randomly rotated matrices and of the compression (upper left corner) of a randomly rotated matrix.