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

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.

25 pages, 8 figures, to appear, Bull. Amer. Math. Soc. Various corrections and referee suggestions incorporated

References in corpus (4)

• Smoothed analysis of algorithms
• A simple invariance theorem
• Universality for mathematical and physical systems
• Circular law, Extreme Singular values and Potential theory