It has been proven that the CDF of the eigenvalue distribution of random Gaussian matrix converges to a uniform disk circular law. Is it true for the PDF of the limiting eigenvalue distribution? In general, convergence in distribution doesn't say anything about the PDF, and I wonder whether it's also the case here. Thanks.

complexmatrix with i.i.d. Gaussian matrix elements; if the matrix elements arereal, then this limit only holds when ${\rm Im}\,\lambda\neq 0$. Some $\sqrt n$ eigenvalues accumulate on the real axis. $\endgroup$