The Elliptic Law

Abstract

A fundamental problem in random matrix theory is to determine the limiting distribution of the ESD as the size of the matrix tends to infinity. In certain cases when the entries have special distribution, such as Gaussian, the joint distribution of the eigenvalues can be given explicitly, and so the limiting distribution can be derived directly. However, these explicit formulas are not available for many random matrix ensembles, and so the problem of finding the limiting distribution becomes much more difficult. On the other hand, the well-known universality phenomenon in random matrix theory predicts that the limiting distribution should not depend on the distribution of the entries. We give two famous examples below. In the 1950s, Wigner studied the limiting ESD for a large class of random Hermitian matrices whose entries on or above the diagonal are independent [52]. In particular, Wigner showed that, under some additional moment