Circular Law

Abstract

This is a famous conjecture that has been open for more than half a century. At present, only some partial answers are known. The conjecture is stated as follows. Suppose that X _n is an n × n matrix with entries x _kj , where {x _kj , k, j = 1, 2, …} forms an infinite double array of iid complex random variables of mean zero and variance one. Using the complex eigenvalues λ₁, λ₂, … , λ> _n of $\frac{1}{{\sqrt n }}X_n$, we can construct a two-dimensional empirical distribution by

$$\mu _n (x,y) = \frac{1}{n}\# \{ i \le n:R(\lambda _k ) \le x,F(\lambda _k ) \le y\} ,$$

which is called the empirical spectral distribution of the matrix $\frac{1}{{\sqrt n }}X_n$.