Transmission-line matrix method

2013-07-10T11:45:49Z

71.43.204.242: /* Basic principle */

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=J. Math. Phys.|volume=3|page=1199|year=1962}}</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>e^{i\theta_k}</math> with ''0 < θk < 2π'' and ''k=1,2,... n''. (In the CSE each of these ''n'' eigenvalues appears twice.) The [[probability density function]] of the phases ''θk'' is 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>
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>

==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=Ukr. Math. J.|volume=37|page=457|year=1985}}</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.<ref>{{cite journal|author=F. Mezzadri|title=How to generate random matrices from the classical compact groups|journal=Notices of the AMS|volume=54|page=592|year=2007|arxiv=math-ph/0609050}}</ref>

==References==
{{reflist}}

== 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]]

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Transmission-line matrix method