The Elliptic Law

TL;DR

This paper generalizes the circular law in random matrix theory by demonstrating that the empirical spectral distribution of scaled random matrices converges to a uniform law on an ellipsoid under broad conditions.

Contribution

It extends the circular law to a broader class of matrices, showing convergence to an ellipsoid law under general assumptions.

Findings

Spectral distribution converges to an ellipsoid law.

Generalizes the circular law to more matrices.

Convergence holds under broad conditions.

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.