Circular law

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 that almost surely (i.e. with probability one), the sequence of measures ${\displaystyle \displaystyle \mu _{{\frac {1}{\sqrt {n}}}X_{n}}}$ converges in distribution to the uniform measure on the unit disk.

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] Guangming Pan and Wang Zhou,[5] and Friedrich Götze and Alexander Tikhomirov.[6] Finally, in 2010 Tao and Vu proved[7] the circular law under the minimal assumptions stated above.

The circular law result was extended in 1988 by Sommers, Crisanti, Sompolinsky and Stein to an elliptical law for ensembles of matrices with arbitrary correlations.[8] The elliptic and circular laws were further generalized by Aceituno, Rogers and Schomerus to the hypotrochoid law which includes higher order correlations.[9]

• Wigner semicircle distribution