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Let me add some remarks in addition to the answer of Carlo, in particular, concerning the numerical side. In free probability terms, the solution is given by the free convolution of the Marchenko-Pas …

If $A_N$ and $B_N$ are two independent Girko ensembles then one should have that they are asymptotically $*$-free; the same should be true if the second ensemble $B_N$ is a sequence of deterministic m …

The following only addresses the last question on the relation between eigenvalues and singular values. What the eigenvalue distribution does not tell you, are traces in products of $M$ and $M^*$; si …

There is the possibility of dealing with any non-commutative rational function in A and B, by using more general (operator-valued) versions of free probability. Usually the corresponding equations can …

Since the two-sided shift is a normal operator this is not really a question about the Brown measure, but about the spectral measure, so it suffices to compute the $*$-moments $\tau(L^nL^{*m})$. For a …

Eigenvalue correlations on the macroscopic level (fluctuations of linear statistics) are in free probability theory captured by the notion of "second order freeness"; a treatment of this can be found …

The formula as stated in your reference seems to be wrong. The Kreweras complement of the partitions should also show up. You can find the correct version in my original paper https://link.springer.co …

Here are a few additional references, in the same directions as in Carlo's answer. Wireless communication: Channel Estimation and Robust Detection for IQ Imbalanced Uplink Massive MIMO-OFDM With Adjus …

Another update: in the paper https://arxiv.org/abs/1608.04923 it is shown how one can use methods from free probability theory to go also beyond the Ginibre ensemble and treat eigenvector correlations …

The $R$-transform is related to harmonic analysis around free products of groups. Actually, the computational machinery for the $R$-transform was also found independently from Voiculescu and about the …

Following up on the answers in the context of free probability theory one should note that there is actually a quite important and canonical operator on a Hilbert space having the semicircle as distri …

I am not sure whether this counts as heuristics, as it goes even deeper into free probability results, but it might give some high-level kind of idea why this result should be true. The main fact abou …

Motivated partly by questions like this I have addressed the question on asymptotic freeness of Wigner matrices in my blog on free probability; I hope to come back with a summary of nice statements an …

Such "concrete" formulas are addressed in Section 3.4. of my memoir Combinatorial Theory of the Free Product with Amalgamation and Operator-Valued Free Probability Theory. This is in the more general …