imported>Brienanni: the correct term is "measures" not "measures", not Undid revision 1271974820 by Jeruain (talk)

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the correct term is "measures" not "measures", not Undid revision 1271974820 by Jeruain (talk)

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In the theory of [[random matrix|random matrices]], the '''circular ensembles''' are measures on spaces of [[unitary matrix|unitary matrices]] introduced by [[Freeman Dyson]] as modifications of the [[Gaussian matrix ensemble]]s.<ref>{{cite journal|author=F.M. Dyson|title=The threefold way. Algebraic structure of symmetry groups and ensembles in quantum mechanics|journal= Journal of Mathematical Physics|volume=3|issue=6|page=1199|year=1962|doi=10.1063/1.1703863|bibcode=1962JMP.....3.1199D}}</ref> The three main examples are the '''circular orthogonal ensemble''' (COE) on symmetric unitary matrices, the '''circular unitary ensemble''' (CUE) on unitary matrices, and the '''circular symplectic ensemble''' (CSE) on self dual unitary quaternionic matrices.

==Probability distributions==

The distribution of the unitary circular ensemble CUE(''n'') is the [[Haar measure]] on the [[unitary group]] ''U(n)''. If ''U'' is a random element of CUE(''n''), then ''UTU'' is a random element of COE(''n''); if ''U'' is a random element of CUE(''2n''), then ''URU'' is a random element of CSE(''n''), where
:<math> U^R = \left( \begin{array}{ccccccc} 0 & -1 & & & & & \\ 1 & 0 & & & & & \\ & & 0 & -1 & & & \\ & & 1 & 0 & & & \\ & & & & \ddots & & \\ & & & & & 0& -1\\ & & & & & 1 & 0 \end{array} \right) U^T \left( \begin{array}{ccccccc} 0 & 1 & & & & & \\ -1 & 0 & & & & & \\ & & 0 & 1 & & & \\ & & -1 & 0 & & & \\ & & & & \ddots & & \\ & & & & & 0& 1\\ & & & & & -1 & 0 \end{array} \right)~. </math>

Each element of a circular ensemble is a unitary matrix, so it has eigenvalues on the unit circle: <math>\lambda_k=e^{i\theta_k}</math> with <math>0 \leq \theta_k < 2\pi</math> for ''k=1,2,... n'', where the <math>\theta_k</math> are also known as '''eigenangles''' or '''eigenphases'''. In the CSE each of these ''n'' eigenvalues appears twice. The distributions have [[probability density function|densities]] with respect to the eigenangles, given by
: <math> p(\theta_1, \cdots, \theta_n) = \frac{1}{Z_{n,\beta}} \prod_{1 \leq k < j \leq n} |e^{i \theta_k} - e^{i \theta_j}|^\beta~</math>
on <math>\R_{[0,2\pi]}^n</math> (symmetrized version), where β=1 for COE, β=2 for CUE, and β=4 for CSE. The normalisation constant ''Zn,β'' is given by
: <math> Z_{n,\beta} = (2\pi)^n \frac{\Gamma(\beta n/2 + 1)}{\left(\Gamma(\beta/2 + 1)\right)^n}~,</math>
as can be verified via [[Selberg integral|Selberg's integral formula]], or Weyl's integral formula for compact Lie groups.

==Generalizations==

Generalizations of the circular ensemble restrict the matrix elements of ''U'' to real numbers [so that ''U'' is in the [[orthogonal group]] ''O(n)''] or to real [[quaternion]] numbers [so that ''U'' is in the [[symplectic group]] ''Sp(2n)''. The Haar measure on the orthogonal group produces the '''circular real ensemble''' (CRE) and the Haar measure on the symplectic group produces the '''circular quaternion ensemble''' (CQE).

The eigenvalues of orthogonal matrices come in complex conjugate pairs <math>e^{i\theta_k}</math> and <math>e^{-i\theta_k}</math>, possibly complemented by eigenvalues fixed at ''+1'' or ''-1''. For ''n=2m'' even and ''det U=1'', there are no fixed eigenvalues and the phases ''θk'' have [[probability distribution]]<ref>{{cite journal|author=V.L. Girko|title=Distribution of eigenvalues and eigenvectors of orthogonal random matrices|journal= Ukrainian Mathematical Journal|volume=37|issue=5|page=457|year=1985|doi=10.1007/bf01061167|s2cid=120597749 }}</ref>
: <math> p(\theta_1, \cdots, \theta_m) = C \prod_{1 \leq k < j \leq m} (\cos\theta_k - \cos\theta_j)^2~,</math>
with ''C'' an unspecified normalization constant. For ''n=2m+1'' odd there is one fixed eigenvalue ''σ=det U'' equal to ±1. The phases have distribution
: <math> p(\theta_1, \cdots, \theta_m) = C \prod_{1 \leq i \leq m}(1-\sigma\cos\theta_i) \prod_{1 \leq k < j \leq m} (\cos\theta_k - \cos\theta_j)^2~.</math>
For ''n=2m+2'' even and ''det U=-1'' there is a pair of eigenvalues fixed at ''+1'' and ''-1'', while the phases have distribution
: <math> p(\theta_1, \cdots, \theta_m) = C \prod_{1 \leq i \leq m}(1-\cos^2\theta_i) \prod_{1 \leq k < j \leq m} (\cos\theta_k - \cos\theta_j)^2~.</math>
This is also the distribution of the eigenvalues of a matrix in ''Sp(2m)''.

These probability density functions are referred to as '''Jacobi distributions''' in the theory of random matrices, because correlation functions can be expressed in terms of [[Jacobi polynomials]].

==Calculations==

Averages of products of matrix elements in the circular ensembles can be calculated using [[Weingarten function]]s. For large dimension of the matrix these calculations become impractical, and a numerical method is advantageous. There exist efficient algorithms to generate random matrices in the circular ensembles, for example by performing a [[QR decomposition]] on a Ginibre matrix.<ref>{{cite journal|author=F. Mezzadri|title=How to generate random matrices from the classical compact groups|url=https://www.ams.org/notices/200705/fea-mezzadri-web.pdf|journal=Notices of the AMS|volume=54|page=592|year=2007|arxiv=math-ph/0609050|bibcode=2006math.ph...9050M}}</ref>

==References==
{{reflist}}

== Software Implementations ==
* {{cite web| url= https://www.wolfram.com/language/11/random-matrices/circular-ensembles-coe-cue-.html| title= Wolfram Mathematica circular ensembles | publisher= [[Wolfram Language]]}}
* {{cite web| title= Bristol: A Python package for Random Matrix Ensembles (Parallel implementation of circular ensemble generation) | date = 2017 | url = https://dx.doi.org/10.5281/zenodo.579642 | doi = 10.5281/zenodo.579642 | last1 = Suezen | first1 = Mehmet }}
** {{cite web | url=https://pypi.org/project/bristol/| title= Bristol: A Python package for Random Matrix Ensembles | publisher= [[pypi]]}}

== External links ==
*{{Citation | last1=Mehta | first1=Madan Lal | title=Random matrices | publisher=Elsevier/Academic Press, Amsterdam | edition=3rd | series=Pure and Applied Mathematics (Amsterdam) | isbn=978-0-12-088409-4 | mr=2129906 | year=2004 | volume=142}}
*{{Citation | last1=Forrester | first1=Peter J. | title=Log-gases and random matrices | publisher=Princeton University Press | isbn=978-0-691-12829-0 | year=2010}}

[[Category:Random matrices]]
[[Category:Mathematical physics]]
[[Category:Freeman Dyson]]

Circular ensemble - Revision history

imported>Brienanni: the correct term is "measures" not "measures", not Undid revision 1271974820 by Jeruain (talk)