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