Thanks to Voiculescu’s freeness, one knows that the normalized eigenvalue counting measure of a selfadjoint non-commutative polynomial in iid GUE’s converges in the  limit of large dimension, and there exist many tools to compute its limiting distribution. On the other hand, on the limiting space (a free product algebra), lots of progress has been made in understanding non-commutative rational fractions. A question by Speicher is whether these rational fractions admit matrix models too. I will explain why the natural candidate is actually a matrix model. In other words, bearing in mind that we already understand the asymptotics of the eigenvalue counting measure of a matrix model obtained as sums, scalings products of iid random matrices, we will show that we can do the same if we allow in addition multiple uses of the matrix inverse when creating our matrix model. 

This is based on arXiv/2103.05962, written in collaboration with Tobias May, Akihiro Miyagawa, Felix Parraud and Sheng Yin.