Circular law

In probability theory, more specifically the study of random matrices, the circular law concerns the distribution of eigenvalues of an $n &times; n$ random matrix with independent and identically distributed entries in the limit $n &rarr; &infin;$.

It asserts that for any sequence of random $n &times; n$ 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.

Ginibre ensembles
The complex Ginibre ensemble is defined as $$X = \frac{1}{\sqrt{2}} Z_1+\frac{i}{\sqrt{2}} Z_2 $$ for $$Z_1, Z_2 \in \R^{n \times n} $$, with all their entries sampled IID from the standard normal distribution $$\mathcal N (0, 1)  $$.

The real Ginibre ensemble is defined as $$X = Z_1$$.

Eigenvalues
The eigenvalues of $$X $$ are distributed according to $$\rho_n\left(z_1, \ldots, z_n\right)=\frac{1}{\pi^n \prod_{k=1}^n k!} \exp \left(-\sum_{k=1}^n\left|z_k\right|^2\right) \prod_{1 \leq j<k \leq n}\left|z_j-z_k\right|^2$$

Global law
Let $$(X_n)_{n=1}^\infty$$ be a sequence sampled from the complex Ginibre ensemble. 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
 * $$ \mu_{\frac{1}{\sqrt{n}} X_n}(A) = n^{-1} \#\{j \leq n : \lambda_j \in A \}~, \quad A \in \mathcal{B}(\mathbb{C}).$$

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

Edge statistics
Let $$G_n$$ be sampled from the real or complex ensemble, and let $$\rho(G_n)$$ be the absolute value of its maximal eigenvalue:$$\rho(G_n) := \max_j |\lambda_j|$$We have the following theorem for the edge statistics:

This theorem refines the circular law of the Ginibre ensemble. In words, the circular law says that the spectrum of $$\frac{1}{\sqrt{n}} G_n$$ almost surely falls uniformly on the unit disc. and the edge statistics theorem states that the radius of the almost-unit-disk is about $$1-\sqrt{\frac{\gamma_n}{4 n}}$$, and fluctuates on a scale of $$\frac{1}{\sqrt{4 n \gamma_n}}$$, according to the Gumbel law.

History
For random matrices with Gaussian distribution of entries (the Ginibre ensembles), the circular law was established in the 1960s by Jean Ginibre. In the 1980s, Vyacheslav Girko introduced an approach which allowed to establish the circular law for more general distributions. Further progress was made 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, Guangming Pan and Wang Zhou, and Friedrich Götze and Alexander Tikhomirov. Finally, in 2010 Tao and Vu proved the circular law under the minimal assumptions stated above.

The circular law result was extended in 1985 by Girko to an elliptical law for ensembles of matrices with a fixed amount of correlation between the entries above and below the diagonal. The elliptic and circular laws were further generalized by Aceituno, Rogers and Schomerus to the hypotrochoid law which includes higher order correlations.