# Circular law

In probability theory, more specifically the study of random matrices, the circular law describes the distribution of eigenvalues of an $n \times n$ random matrix with independent and identically distributed entries in the limit $n \to \infty$.

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.

## Precise statement

Let $(X_n)_{n=1}^\infty$ be a sequence of $n \times n$ 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 $\displaystyle \mu_{\frac{1}{\sqrt{n}} X_n}(x,y)$ converges weakly 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 1960-s 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.

## References

1. ^ Ginibre, Jean (1965). "Statistical ensembles of complex, quaternion, and real matrices". J. Mathematical Phys. 6: 440–449. doi:10.1063/1.1704292.
2. ^ Girko, V.L. (1984). "The circular law". Teor. Veroyatnost. i Primenen. 29 (4): 669–679.
3. ^ Bai, Z.D. (1997). "Circular law". Ann. Probab. 25 (1): 494–529. doi:10.1214/aop/1024404298.
4. ^ Tao, T.; Vu, V.H. (2008). "Random matrices: the circular law.". Commun. Contemp. Math. 10 (2): 261–307.
5. ^ Götze, F.; Tikhomirov, A. (2010). "The circular law for random matrices". Ann. Probab. 38 (4): 1444–1491.
6. ^ Tao, Terence; Vu, Van (2010), "Random matrices: Universality of ESD and the Circular Law (with appendix by M. Krishnapur)", Annals of Probability 38 (5): 2023–2065, arXiv:0807.4898, doi:10.1214/10-AOP534, MR2722794

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