Lecture Notes on Free Probability

Abstract

These lecture notes provide an introduction to free probability theory, with a focus on tools and techniques useful in the study of large random matrices. Topics include freeness, free cumulants, additive and multiplicative free convolution, the R- and S-transforms, subordination theory, and operator-valued extensions. Applications to asymptotic freeness and linearization methods are discussed in detail. The notes aim to be accessible to graduate students with a background in functional analysis and probability. The lecture notes were originally written for a graduate course. They are updated to include recent results on subordination and linearization methods in free probability.