Another important random matrix ensemble is given by Haar unitary random matrices – these are unitary matrices equipped with the Haar measure as corresponding probability measure. We will see that one can get asymptotic freeness results for Haar unitary random matrices similar to those we derived for Gaussian random matrices in the last lecture. We will also see that we have asymptotic freeness between constant matrices which are randomly rotated by a Haar unitary random matrix. (This will follow from the fact that conjugation by a free Haar unitary can be used to make general random variables free.)

Our calculations for the unitary random matrices will be of a similar kind to those from the last lecture. The main ingredient is a Wick type formula for correlations of the entries of the Haar unitary random matrices.

Haar unitary random matrices

Remark 23.1. A fundamental fact in abstract harmonic analysis is that any compact group has an analog of the Lebesgue measure, the so-called Haar measure, which is characterized by the fact that it is invariant under translations by group elements. This Haar measure is finite and unique up to multiplication with a constant, thus we can normalize it to a probability measure – the unique Haar probability measure on the compact group.