Numerical Free Probability: Computing Eigenvalue Distributions of Algebraic Manipulations of Random Matrices
by Sheehan Olver
Abstract: Suppose that the global eigenvalue density of two large random matrix ensembles A and B are known. It is a remarkable fact that, under broad conditions, the eigenvalue density of A + B and (if A and B are positive definite) A*B are uniquely determined from only the eigenvalue densities of A and B; i.e., no information about eigenvectors are required. These operations on eigenvalue distributions are described by free probability theory. We construct a numerical toolbox that can efficiently and reliably calculate these operations with spectral accuracy, by exploiting the complex analytical framework that underlies free probability theory.