Free probability theory

This article focuses on free probability theory, which is useful for dealing with asymptotic eigenvalue distributions in situations involving several matrices. In particular, it considers some of the basic ideas and results of free probability theory, mostly from the random matrix perspective. After providing a brief background on free probability theory, the article discusses the moment method for several random matrices and the concept of freeness. It then gives some of the main probabilistic notions used in free probability and introduces the combinatorial theory of freeness. In this theory, freeness is described in terms of free cumulants in relation to the planar approximations in random matrix theory (RMT). The article also examines free harmonic analysis, second-order freeness, operator-valued free probability theory, further free-probabilistic aspects of random matrices, and operator algebraic aspects of free probability.