circular law

The complex Ginibre ensemble is defined as $X\; =\; \backslash frac\{1\}\{\backslash sqrt\{2\}\}\; Z\_1+\backslash frac\{i\}\{\backslash sqrt\{2\}\}\; Z\_2$ for $Z\_1,\; Z\_2\; \backslash in\; \backslash R^\{n\; \backslash times\; n\}$, with all their entries sampled IID from the standard normal distribution $\backslash mathcal\; N\; (0,\; 1)$.

The real Ginibre ensemble is defined as $X\; =\; Z\_1$.

The eigenvalues of $X$ are distributed according to{{cite arXiv |last=Meckes |first=Elizabeth |title=The Eigenvalues of Random Matrices |date=2021-01-08 |class=math.PR |eprint=2101.02928}}

Let $(X\_n)\_\{n=1\}^\backslash infty$ be a sequence sampled from the complex Ginibre ensemble. Let $\backslash lambda\_1,\; \backslash ldots,\; \backslash lambda\_n,\; 1\; \backslash leq\; j\; \backslash leq\; n$ denote the eigenvalues of $\backslash frac\{1\}\{\backslash sqrt\{n\}\}X\_n$. Define the empirical spectral measure of $\backslash displaystyle\; \backslash frac\{1\}\{\backslash sqrt\{n\}\}\; X\_n$ as

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

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

Let $G\_n$ be sampled from the real or complex ensemble, and let $\backslash rho(G\_n)$ be the absolute value of its maximal eigenvalue:$$\backslash rho(G\_n)\; :=\; \backslash max\_j\; |\backslash lambda\_j|$$We have the following theorem for the edge statistics:{{Cite journal |last=Rider |first=B |date=2003-03-28 |title=A limit theorem at the edge of a non-Hermitian random matrix ensemble |url=https://iopscience.iop.org/article/10.1088/0305-4470/36/12/331 |journal=Journal of Physics A: Mathematical and General |volume=36 |issue=12 |pages=3401–3409 |doi=10.1088/0305-4470/36/12/331 |bibcode=2003JPhA...36.3401R |issn=0305-4470|url-access=subscription }}{{Math theorem

| math_statement = For $G\_n$ and $\backslash rho\backslash left(G\_n\backslash right)$ as above, with probability one,

$$\backslash lim\; \_\{n\; \backslash rightarrow\; \backslash infty\}\; \backslash frac\{1\}\{\backslash sqrt\{n\}\}\; \backslash rho\backslash left(G\_n\backslash right)=1$$

Moreover, if $\backslash gamma\_n=\backslash log\; \backslash left(\backslash frac\{n\}\{2\; \backslash pi\}\backslash right)-2\; \backslash log\; (\backslash log\; (n))$ and

$$Y\_n:=\backslash sqrt\{4\; n\; \backslash gamma\_n\}\backslash left(\backslash frac\{1\}\{\backslash sqrt\{n\}\}\; \backslash rho\backslash left(G\_n\backslash right)-1-\backslash sqrt\{\backslash frac\{\backslash gamma\_n\}\{4\; n\}\}\backslash right),$$

then $Y\_n$ converges in distribution to the Gumbel law, i.e., the probability measure on $\backslash mathbb\{R\}$ with cumulative distribution function $F\_\{\backslash mathrm\{Gum\}\}(x)=e^\{-e^\{-x\}\}$.

| name = Edge statistics of the Ginibre ensemble

| note =

}}

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

For random matrices with Gaussian distribution of entries (the Ginibre ensembles), the circular law was established in the 1960s by Jean Ginibre.{{cite journal|last=Ginibre|first=Jean|title=Statistical ensembles of complex, quaternion, and real matrices|journal=J. Math. Phys.|year=1965|volume=6|issue=3 |pages=440–449|doi=10.1063/1.1704292|mr=0173726|bibcode=1965JMP.....6..440G}} In the 1980s, Vyacheslav Girko introduced{{cite journal|last=Girko|first=V.L.|title=The circular law|journal=Teoriya Veroyatnostei i ee Primeneniya|year=1984|volume=29|issue=4|pages=669–679}} an approach which allowed to establish the circular law for more general distributions. Further progress was made{{cite journal|last=Bai|first=Z.D.|title=Circular law|journal=Annals of Probability|year=1997|volume=25|issue=1|pages=494–529|doi=10.1214/aop/1024404298|mr=1428519|doi-access=free}} 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,{{cite journal|last1=Tao|first1=T.|last2=Vu|first2=V.H.|title=Random matrices: the circular law.|journal=Commun. Contemp. Math.|year=2008|volume=10|issue=2|pages=261–307|doi=10.1142/s0219199708002788|mr=2409368|arxiv=0708.2895|s2cid=15888373 }} Guangming Pan and Wang Zhou,{{cite journal|last1=Pan|first1=G.|last2=Zhou|first2=W.|title=Circular law, extreme singular values and potential theory.|journal=J. Multivariate Anal.|year=2010|volume=101|issue=3|pages=645–656|doi=10.1016/j.jmva.2009.08.005|arxiv=0705.3773|s2cid=7475359 }} and Friedrich Götze and Alexander Tikhomirov.{{cite journal|last1=Götze|first1=F.|last2=Tikhomirov|first2=A.|title=The circular law for random matrices|journal=Annals of Probability|year=2010|volume=38|issue=4|pages=1444–1491|doi=10.1214/09-aop522|mr=2663633|arxiv=0709.3995|s2cid=1290255 }} Finally, in 2010 Tao and Vu proved{{cite journal|last1=Tao|first1=Terence|author1-link=Terence Tao|last2=Vu|first2=Van|author2-link=Van Vu|title=Random matrices: Universality of ESD and the Circular Law|others=appendix by Manjunath Krishnapur|journal=Annals of Probability|volume=38|issue=5|year=2010|pages=2023–2065|arxiv=0807.4898|doi=10.1214/10-AOP534|mr=2722794|s2cid=15769353 }} the circular law under the minimal assumptions stated above.

The circular law result was extended in 1985 by Girko{{cite journal|last=Girko|first=V.L.|title=The elliptic law|journal=Teoriya Veroyatnostei i ee Primeneniya|year=1985|volume=30|pages=640–651}} 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.{{cite journal|last1=Aceituno|first1=P.V.|last2=Rogers|first2=T.|last3=Schomerus|first3=H.|title=Universal hypotrochoidic law for random matrices with cyclic correlations.|journal=Physical Review E|year=2019|volume=100|issue=1|pages=010302|doi=10.1103/PhysRevE.100.010302 |pmid=31499759 |arxiv=1812.07055 |bibcode=2019PhRvE.100a0302A |s2cid=119325369 }}

• Wigner semicircle distribution

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Category:Random matrices