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Let $X_{N}\in\mathcal{M}_{N}\big(L^{\infty-}(\Omega,\mathbb{P})\big)$ be a $N\times N$ random complex matrix such its entries $(x_{ij}, 1\leq i, j\leq N)$ be $i.i.d.$, centred with variance $1$. $X_{N}$ is called a non-selfadjoint random matrices. In particular, if the entries follow a Gaussian law, the matrix is said non-selfadjoin Gaussian random matrix.

Whereas non-selfadjoint Gaussian random matrices are unitarily invariant this is not true any more for general non-Hermitian random matrices; thus we cannot use the results about Haar unitary random matrices or Wick formula to derive asymptotic freeness results for general non-selfadjoint random matrices. Nevertheless, there are many results in the literature which show that non-Hermitian random matrices 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 circle, $i.e$,

$ \mu_{\frac{1}{\sqrt{N}}X_{N}}=\frac{1}{N}\#\lbrace i\leq N:\text{Re }(\lambda_{i})\leq x,\ \text{Im}(\lambda_{i})\leq y \rbrace $ converge in distribution to the circular law $ \mu_{circ}=\frac{1}{\pi} 1_{\mid x^{2}\mid+ \mid y^{2} \mid \leq 1},$ where the $\lambda_{i}$ are the eigenvalues of $ \mu_{\frac{1}{\sqrt{N}}X_{N}}$, and $x,y$ are the real and imaginary coordinates of the complex plane.

Let $G_N$, $G'_N$ be two indépendent non-selfadjoint Gaussian random matrices. By Girko Theorem, $G_N$, $G'_N$ converge in distribution to circular elements $\mu_{circ}, \mu'_{circ}$, if $N\to\infty$. In Voiculescu's Free Probability, this means that

$G_N\xrightarrow[N\to+\infty]{}c$, $G'_N\xrightarrow[N\to+\infty]{}c'$,

where $c$, $c'$ are two circular elements. Then, $G_N$, $G'_N$ are asymptotically free, $i.e,$

$(G_N, G'_N)\xrightarrow[N\to+\infty]{}(c, c')$ in distribution, and $(c, c')$ are free.

How about the asymptotic freeness of general two non-selfadjoint random matrices $X_N, X'_N$?

Is it true that $X_N, X'_N$ converge in distribution to two free circular elements, when $N\to\infty$ like in the Gaussian case bellow?

One can ask the same questions for sample covariance matrices, whih are general Wishart matrices.

I know that the similar problem in the context of selfadjoint random matrices - the so called Wigner matrices- is solved by respectively Dykema; Anderson, Guionet and Zeitouni; Speicher and Mingo. Wigner matrices are random matrices whcih are random matrices with the entries are, apart from symmetry conditions, independent and identically distributed but with arbitrary, not necessarily Gaussian, distribution.

In one of his articles, Prof. Roland Speicher Wrote: "In general, the distribution of $p(X_N, Y_N)$ will depend on the relation between the eigenspaces of $X_N$ and of $Y_N$. However, by the concentration of measure phenomenon, we expect that for large N this relation between the eigenspaces concentrates on typical or generic positions, and that then the asymptotic eigenvalue distribution of $p(X_N, Y_N)$ depends in a deterministic way only on the asymptotic eigenvalue distribution of $X_N$ and on the asymptotic eigenvalue distribution of $Y_N.$", where here $p(X,N, Y_N)$ is a non commutative polynomials in two selfadjoint Gaussian random matrices. What does intuitively typical or generic positions mean?

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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 and clean proofs sometimes soon ...

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