On the concentration of random multilinear forms and the universality of random block matrices

TL;DR

This paper extends the circular law to a broad class of random block matrices with dependent entries, using inverse concentration results for multilinear forms to establish spectral distribution convergence.

Contribution

It introduces a novel approach to prove the circular law for dependent block matrices via inverse concentration inequalities.

Findings

Circular law holds for certain dependent block matrices

Spectral distribution converges to uniform on the unit disk

New inverse concentration results for multilinear forms

Abstract

The circular law asserts that if $X_n$ is a $n \times n$ matrix with iid complex entries of mean zero and unit variance, then the empirical spectral distribution of $\frac{1}{\sqrt{n}} X_n$ converges almost surely to the uniform distribution on the unit disk as $n$ tends to infinity. Answering a question of Tao, we prove the circular law for a general class of random block matrices with dependent entries. The proof relies on an inverse-type result for the concentration of linear operators and multilinear forms.