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
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“The source of Pyrrhonism comes from failing to distinguish between a demonstration, a proof and a probability. A demonstration supposes that the contradictory idea is impossible; a proof of fact is where all the reasons lead to belief, without there being any pretext for doubt; a probability is where the reasons for belief are stronger than those for doubting.”
—Andrew Michael Ramsay (1686–1743)