# Circular law

In probability theory, more specifically the study of random matrices, the circular law is the distribution of eigenvalues of an n × n random matrix with independent and identically distributed entries in the limit n → ∞.

It asserts that for any sequence of random n × n matrices whose entries are independent and identically distributed random variables, all with mean zero and variance equal to 1/n, the limiting spectral distribution is the uniform distribution over the unit disc.

Plot of the real and imaginary parts (scaled by sqrt(1000)) of the eigenvalues of a 1000x1000 matrix with independent,standard normal entries.

## Precise statement

Let ${\displaystyle (X_{n})_{n=1}^{\infty }}$ be a sequence of n × n matrix ensembles whose entries are i.i.d. copies of a complex random variable x with mean 0 and variance 1. Let ${\displaystyle \lambda _{1},\ldots ,\lambda _{n},1\leq j\leq n}$ denote the eigenvalues of ${\displaystyle \displaystyle {\frac {1}{\sqrt {n}}}X_{n}}$. Define the empirical spectral measure of ${\displaystyle \displaystyle {\frac {1}{\sqrt {n}}}X_{n}}$ as

${\displaystyle \mu _{{\frac {1}{\sqrt {n}}}X_{n}}(A)=n^{-1}\#\{j\leq n:\lambda _{j}\in A\}~,\quad A\in {\mathcal {B}}(\mathbb {C} ).}$

With these definitions in mind, the circular law asserts the sequence of measures ${\displaystyle \displaystyle \mu _{{\frac {1}{\sqrt {n}}}X_{n}}(x,y)}$ converges in distribution to the uniform measure on the unit disk.

## History

For random matrices with Gaussian distribution of entries (the Ginibre ensembles), the circular law was established in the 1960s by Jean Ginibre.[1] In the 1980s, Vyacheslav Girko introduced[2] an approach which allowed to establish the circular law for more general distributions. Further progress was made[3] by Zhidong Bai, who established the circular law under certain smoothness assumptions on the distribution.

The assumptions were further relaxed in the works of Terence Tao and Van H. Vu,[4] Friedrich Götze and Alexander Tikhomirov.[5] Finally, in 2010 Tao and Vu proved[6] the circular law under the minimal assumptions stated above.