The Elliptic Law

Abstract

We show that, under some general assumptions on the entries of a random complex $n\times n$ matrix $X_n$, the empirical spectral distribution of $\frac{1}{\sqrt{n}}{X}_n$ converges to the uniform law of an ellipsoid as $n$ tends to infinity. This generalizes the well-known circular law in random matrix theory.

57 pages; minor corrections