From the Littlewood-Offord problem to the Circular Law: universality of the spectral distribution of random matrices

TL;DR

This paper surveys recent progress in establishing the circular law for random matrices, highlighting advances in understanding the Littlewood-Offord problem and its role in spectral distribution universality.

Contribution

It reviews key methods and recent developments that proved the circular law under minimal assumptions on matrix entries, emphasizing the Littlewood-Offord problem's significance.

Findings

Proof of the circular law for a broad class of random matrices

Advances in inverse Littlewood-Offord problem understanding

Connection between combinatorial probability and spectral universality

Abstract

The famous \emph{circular law} asserts that if $M_n$ is an $n \times n$ matrix with iid complex entries of mean zero and unit variance, then the empirical spectral distribution (ESD) of the normalized matrix $\frac{1}{\sqrt{n}} M_n$ converges almost surely to the uniform distribution on the unit disk $\{z \in \C: |z| \leq 1 \}$. After a long sequence of partial results that verified this law under additional assumptions on the distribution of the entries, the full circular law was recently established in \cite{TVcir2}. In this survey we describe some of the key ingredients used in the establishment of the circular law, in particular recent advances in understanding the Littlewood-Offord problem and its inverse.