Applications of Realizations (aka Linearizations) to Free Probability

We show how the combination of new "linearization" ideas in free probability theory with the powerful "realization" machinery -- developed over the last 50 years in fields including systems engineering and automata theory -- allows solving the problem of determining the eigenvalue distribution (or even the Brown measure, in the non-selfadjoint case) of noncommutative rational functions of random matrices when their size tends to infinity. Along the way we extend evaluations of noncommutative rational expressions from matrices to stably finite algebras, e.g. type II$_1$ von Neumann algebras, with a precise control of the domains of the rational expressions. The paper provides sufficient background information, with the intention that it should be accessible both to functional analysts and to algebraists.