  We investigate the implications of free probability for random matrices. From
rules for calculating all possible joint moments of two free random matrices,
we develop a notion of partial freeness which is quantified by the breakdown of
these rules. We provide a combinatorial interpretation for partial freeness as
the presence of closed paths in Hilbert space defined by particular joint
moments. We also discuss how asymptotic moment expansions provide an error term
on the density of states. We present MATLAB code for the calculation of moments
and free cumulants of arbitrary random matrices.
