# Circular law

In probability theory, more specifically the study of random matrices, the circular law describes 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.

## Precise statement

Let $(X_n)_{n=1}^\infty$ 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 $\lambda_1, \ldots, \lambda_n, 1 \leq j \leq n$ denote the eigenvalues of $\displaystyle \frac{1}{\sqrt{n}}X_n$. Define the empirical spectral measure of $\displaystyle \frac{1}{\sqrt{n}} X_n$ 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 $\displaystyle \mu_{\frac{1}{\sqrt{n}} X_n}(x,y)$ almost surely converges weakly 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.