the same limiting distribution for the real Ginibre ensemble.

For the general (non-Gaussian) case, there is no formula for the joint distribution of the eigenvalues and the problem appears much more difficult. The universality phenomenon in random matrix theory asserts that the spectral behavior of an iid random matrix does not depend on the distribution of the atom variable $\xi$ in the limit $N \rightarrow \infty$. In other words, one expects that the circular law describes the limiting ESD of a large class of random matrices (not just Gaussian matrices).

An important result was obtained by Girko [@G1; @G2] who related the empirical spectral measure of a non-Hermitian matrix to that of a family of Hermitian matrices. Using this Hermitization technique, Bai [@Bcirc; @BSbook] gave the first rigorous proof of the circular law for general (non-Gaussian) distributions. He proved the result under a number of moment and smoothness assumptions on the atom variable $\xi$, and a series of recent improvements were obtained by Götze and Tikhomirov [@GTcirc], Pan and Zhou [@PZ] and Tao and Vu [@TVcirc; @TVesd]. In particular, Tao and Vu [@TVbull; @TVesd] established the law with the minimum assumption that $\xi$ has finite variance.

\[thm:tvcirc\] Let $\xi$ be 