Free Probability, Random Matrices, and Representations of Non-commutative Rational Functions

Abstract

A fundamental problem in free probability theory is to understand distributions of “non-commutative functions” in freely independent variables. Due to the asymptotic freeness phenomenon, which occurs for many types of independent random matrices, such distributions can describe the asymptotic eigenvalue distribution of corresponding random matrix models when their dimension tends to infinity. For non-commutative polynomials and rational functions, an algorithmic solution to this problem is presented. It relies on suitable representations for these functions.