Suppose we have a set of freely independent random matrices $X_i$ $i=\{1,\dots,n\}$ of dimension $M\times N$. Each $X_i$ has independent and identically distributed entries. Then we make the sum $X=\sum X_iX_i^H$.
From Marchenko–Pastur we know that the empirical distribution of eigenvalues of $X_iX_i^H$ have the Marchenko–Pastur distribution.
Since $X_i$ are all independent can we take the empirical distribution of the eigenvalues of $X$ is the free convolution of the empirical eigenvalue distributions of $X_iX_i^H$s? The confusion is usually individual eigenvalues have no relation to the eigenvalues of the sum matrix.
P.S.: Related to https://mathoverflow.net/questions/155346/random-matrix-determinant-problem/155399?noredirect=1#comment397874_155399.