Tuples of noncommutative random variables that are subjected to conditions such as finiteness of the free Fisher information or maximality of the free entropy dimension are expected to produce “regular” spectral distributions under evaluation of suitable noncommutative functions. In the last years, this intuition was confirmed in various cases. The regularity properties that were addressed range from absence of atoms to Hölder continuity and absolute continuity; the considered types of functions include (matrices of) noncommutative polynomials and rational functions, but also operator-valued elements were studied in great detail. In many of those cases, one observes a strong interplay between the underlying operators on the analytic side and the considered functions on the algebraic side.
In my talk, which is based on joint work with Marwa Banna, Roland Speicher, and Sheng Yin, I will report on some of these results and their applications in random matrix theory.