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
Each element of a circular ensemble is a unitary matrix, so it has eigenvalues on the unit circle: 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
where β=1 for COE, β=2 for CUE, and β=4 for CSE. The normalisation constant Zn,β is given by
Read more about this topic: Circular Ensemble
Famous quotes containing the word probability:
“Crushed to earth and rising again is an authors gymnastic. Once he fails to struggle to his feet and grab his pen, he will contemplate a fact he should never permit himself to face: that in all probability books have been written, are being written, will be written, better than anything he has done, is doing, or will do.”
—Fannie Hurst (18891968)