Free Probability Theory

Now we come to one of the most important and inspiring realizations of freeness. Up to now we have realized free random variables by generators of the free group factors or by creation and annihilation operators on full Fock spaces. This was not very surprising because our definition of freeness was modelled according to such situations. But there are objects from a quite different mathematical universe which are also free (at least asymptotically) namely special random matrices. Random matrices are matrices whose entries are classical random variables, and the most important class of random matrices are the so-called Gaussian random matrices where the entries are classical Gaussian random variables. So, before we talk about random matrices, we should recall the basic properties of Gaussian random variables.