Circular Ensemble - Probability Distributions

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 < 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

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