Applications of Realizations (aka Linearizations) to Free Probability

Understand

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.