Circular ensemble について

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Circular ensemble ：ウィキペディア英語版

Circular ensemble
In the theory of random matrices, the circular ensembles are measures on spaces of unitary matrices introduced by Freeman Dyson as modifications of the Gaussian matrix ensembles. 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 ''U^TU'' is a random element of COE(''n''); if ''U'' is a random element of CUE(''2n''), then ''U^RU'' is a random element of CSE(''n''), where
:

U^R = \left( \begin 0 & -1 & & & & &  \\ 1 & 0 &  & & & & \\ & & 0 & -1 &  & &  \\ & & 1 & 0  & & & \\ & & & & \ddots & & \\ & & & & & 0& -1\\ & & & & & 1 & 0 \end \right) U^T \left( \begin 0 & 1 & & & & &  \\ -1 & 0 &  & & & & \\ & & 0 & 1 &  & &  \\ & & -1 & 0  & & & \\ & & & & \ddots & & \\ & & & & & 0& 1\\ & & & & & -1 & 0 \end \right)~.

Each element of a circular ensemble is a unitary matrix, so it has eigenvalues on the unit circle:

e^

with ''0 < θ_k < 2π'' and ''k=1,2,... n''. (In the CSE each of these ''n'' eigenvalues appears twice.) The probability density function of the phases ''θ_k'' is given by
:

p(\theta_1, \cdots, \theta_n) = \frac |e^ - e^|^\beta~,

where β=1 for COE, β=2 for CUE, and β=4 for CSE. The normalisation constant ''Z_n,β'' is given by
:

Z_ = (2\pi)^n \frac~.

抄文引用元・出典: フリー百科事典『ウィキペディア（Wikipedia）』
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