# Are they the $N\times N$ random matrices with real or complex entries asymptotically free?

The $N\times N$ random matrices with real or complex entries are generalizations of non-hermitian gaussian ensembles, also known as Girko ensemble: the entries are independent and identically distributed, but with arbitrary, not necessarily Gaussian, distribution. Whereas non-hermitian gaussian ensembles are unitarily invariant this is not true any more in the general case; thus we cannot use the results about Haar unitary random matrices to derive asymptotic freeness results for the non-hermitian and non gaussian ensembles. Nevertheless, there are many results in the literature which show that the $N\times N$ random matrices with real or complex behave with respect to eigenvalue questions in the same way as non-hermitian Gaussian random matrices. For example, their eigenvalue distribution converges always to a circular law. Are they the $N\times N$ random matrices with real or complex entries asymptotically free?

• asymptotic freedom means that the large-$N$ eigenvalue distribution of the sum $A+UBU^\ast$, with $U$ an arbitrary unitary matrix, depends only the large-$N$ eigenvalue distributions of the matrices $A$ and $B$. This is not possible if the ensemble lacks unitary invariance. Jul 1 '18 at 14:24

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 matrices with a limiting $*$-distribution.