Free probability theory is a framework obtained by replacing the notion of independence in classical probability theory with a concept called free independence. The random variables considered in free probability appear as operators on a Hilbert space and possess a noncommutative structure.
On the other hand, a random matrix is a matrix whose entries are random variables, and random matrix theory has found applications not only in mathematics but also in various fields such as quantum physics and machine learning. The relationship between free probability and random matrix theory has been actively studied since the 1990s, following Voiculescu’s discovery of asymptotic freeness.
In this talk, I will introduce several results concerning the relationship between free probability and random matrix theory, accompanied by figures obtained through numerical computations. Finally, I will discuss some of my recent work on the spectra of polynomials in freely independent semicircular and circular distributions.