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

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.

43 pages, 2 figures; simplified the presentation and incorporated the referee's suggestions. To appear in Probability Theory and Related Fields