Second order freeness and fluctuations of random matrices. III: Higher order freeness and free cumulants

Abstract

We extend the relation between random matrices and free probability theory from the level of expectations to the level of all correlation functions (which are classical cumulants of traces of products of the matrices). We introduce the notion of “higher order freeness” and develop a theory of corresponding free cumulants. We show that two independent random matrix ensembles are free of arbitrary order if one of them is unitarily invariant. We prove $\mathrm{R}$-transform formulas for second order freeness. Much of the presented theory relies on a detailed study of the properties of “partitioned permutations”.