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.

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

Each element of a circular ensemble is a unitary matrix, so it has eigenvalues on the unit circle: ${\displaystyle e^{i\theta _{k}}}$ 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

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).

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