Circular law

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

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.

Ginibre ensembles

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

The real Ginibre ensemble is defined as ${\displaystyle X=Z_{1}}$.

Eigenvalues

The eigenvalues of ${\displaystyle X}$ are distributed according to^[1]$${\displaystyle \rho _{n}\left(z_{1},\ldots ,z_{n}\right)={\frac {1}{\pi ^{n}\prod _{k=1}^{n}k!}}\exp \left(-\sum _{k=1}^{n}\left|z_{k}\right|^{2}\right)\prod _{1\leq j<k\leq n}\left|z_{j}-z_{k}\right|^{2}}$$

Plot of the real and imaginary parts (scaled by sqrt(1000)) of the eigenvalues of a 1000x1000 matrix with independent, standard normal entries.

Global law

Let ${\displaystyle (X_{n})_{n=1}^{\infty }}$ be a sequence sampled from the complex Ginibre ensemble. Let ${\displaystyle \lambda _{1},\ldots ,\lambda _{n},1\leq j\leq n}$ denote the eigenvalues of ${\displaystyle {\frac {1}{\sqrt {n}}}X_{n}}$. Define the empirical spectral measure of ${\displaystyle \displaystyle {\frac {1}{\sqrt {n}}}X_{n}}$ as

${\displaystyle \mu _{{\frac {1}{\sqrt {n}}}X_{n}}(A)=n^{-1}\#\{j\leq n:\lambda _{j}\in A\}~,\quad A\in {\mathcal {B}}(\mathbb {C} ).}$

Then, 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.

Edge statistics

Let ${\displaystyle G_{n}}$ be sampled from the real or complex ensemble, and let ${\displaystyle \rho (G_{n})}$ be the absolute value of its maximal eigenvalue:$${\displaystyle \rho (G_{n}):=\max _{j}|\lambda _{j}|}$$We have the following theorem for the edge statistics:^[2]

Edge statistics of the Ginibre ensemble—For ${\displaystyle G_{n}}$ and ${\displaystyle \rho \left(G_{n}\right)}$ as above, with probability one, $${\displaystyle \lim _{n\rightarrow \infty }{\frac {1}{\sqrt {n}}}\rho \left(G_{n}\right)=1}$$

Moreover, if ${\displaystyle \gamma _{n}=\log \left({\frac {n}{2\pi }}\right)-2\log(\log(n))}$ and $${\displaystyle Y_{n}:={\sqrt {4n\gamma _{n}}}\left({\frac {1}{\sqrt {n}}}\rho \left(G_{n}\right)-1-{\sqrt {\frac {\gamma _{n}}{4n}}}\right),}$$ then ${\displaystyle Y_{n}}$ converges in distribution to the Gumbel law, i.e., the probability measure on ${\displaystyle \mathbb {R} }$ with cumulative distribution function ${\displaystyle F_{\mathrm {Gum} }(x)=e^{-e^{-x}}}$.

This theorem refines the circular law of the Ginibre ensemble. In words, the circular law says that the spectrum of ${\displaystyle {\frac {1}{\sqrt {n}}}G_{n}}$ almost surely falls uniformly on the unit disc. and the edge statistics theorem states that the radius of the almost-unit-disk is about ${\displaystyle 1-{\sqrt {\frac {\gamma _{n}}{4n}}}}$, and fluctuates on a scale of ${\displaystyle {\frac {1}{\sqrt {4n\gamma _{n}}}}}$, 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.^[3] In the 1980s, Vyacheslav Girko introduced^[4] an approach which allowed to establish the circular law for more general distributions. Further progress was made^[5] 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,^[6] Guangming Pan and Wang Zhou,^[7] and Friedrich Götze and Alexander Tikhomirov.^[8] Finally, in 2010 Tao and Vu proved^[9] the circular law under the minimal assumptions stated above.

The circular law result was extended in 1985 by Girko^[10] 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.^[11]

See also

• Wigner semicircle distribution

References

1. Meckes, Elizabeth (2021-01-08). "The Eigenvalues of Random Matrices". arXiv:2101.02928 [math.PR].
2. Rider, B (2003-03-28). "A limit theorem at the edge of a non-Hermitian random matrix ensemble". Journal of Physics A: Mathematical and General. 36 (12): 3401–3409. Bibcode:2003JPhA...36.3401R. doi:10.1088/0305-4470/36/12/331. ISSN 0305-4470.
3. Ginibre, Jean (1965). "Statistical ensembles of complex, quaternion, and real matrices". J. Math. Phys. 6 (3): 440–449. Bibcode:1965JMP.....6..440G. doi:10.1063/1.1704292. MR 0173726.
4. Girko, V.L. (1984). "The circular law". Teoriya Veroyatnostei i ee Primeneniya. 29 (4): 669–679.
5. Bai, Z.D. (1997). "Circular law". Annals of Probability. 25 (1): 494–529. doi:10.1214/aop/1024404298. MR 1428519.
6. Tao, T.; Vu, V.H. (2008). "Random matrices: the circular law". Commun. Contemp. Math. 10 (2): 261–307. arXiv:0708.2895. doi:10.1142/s0219199708002788. MR 2409368. S2CID 15888373.
7. Pan, G.; Zhou, W. (2010). "Circular law, extreme singular values and potential theory". J. Multivariate Anal. 101 (3): 645–656. arXiv:0705.3773. doi:10.1016/j.jmva.2009.08.005. S2CID 7475359.
8. Götze, F.; Tikhomirov, A. (2010). "The circular law for random matrices". Annals of Probability. 38 (4): 1444–1491. arXiv:0709.3995. doi:10.1214/09-aop522. MR 2663633. S2CID 1290255.
9. Tao, Terence; Vu, Van (2010). "Random matrices: Universality of ESD and the Circular Law". Annals of Probability. 38 (5). appendix by Manjunath Krishnapur: 2023–2065. arXiv:0807.4898. doi:10.1214/10-AOP534. MR 2722794. S2CID 15769353.
10. Girko, V.L. (1985). "The elliptic law". Teoriya Veroyatnostei i ee Primeneniya. 30: 640–651.
11. Aceituno, P.V.; Rogers, T.; Schomerus, H. (2019). "Universal hypotrochoidic law for random matrices with cyclic correlations". Physical Review E. 100 (1): 010302. arXiv:1812.07055. Bibcode:2019PhRvE.100a0302A. doi:10.1103/PhysRevE.100.010302. PMID 31499759. S2CID 119325369.