Free multiplicative convolution

Free multiplicative convolution is a mathematical operation used in free probability theory to combine two random variables (or distributions) in a way that models their joint behavior when they are "free" or non-commutative, similar to independence in classical probability. Unlike traditional multiplication, it accounts for complex interactions in large random matrices or non-commutative settings, capturing how their spectral distributions (eigenvalues) merge. Essentially, it provides a way to understand the combined spectral behavior of large, non-commuting systems, generalizing classical convolution to the non-commutative realm.