Free convolution

Free convolution is a mathematical concept used in free probability theory to describe how the combined behavior of two random variables works when they are "free" or non-commuting, analogous to independent variables in classical probability. Unlike traditional addition, where probabilities combine straightforwardly, free convolution accounts for non-commutative structures common in quantum physics and certain matrix models. It provides a systematic way to determine the distribution of the sum of free random variables, helping researchers understand complex systems where traditional probability rules don't apply.