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 normalisation constant Zn,β is given by as can be verified via Selberg's integral formula, or Weyl's integral formula for compact Lie groups.
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).
The eigenvalues of orthogonal matrices come in complex conjugate pairs
For n=2m even and det U=1, there are no fixed eigenvalues and the phases θk have probability distribution[2] with C an unspecified normalization constant.
These probability density functions are referred to as Jacobi distributions in the theory of random matrices, because correlation functions can be expressed in terms of Jacobi polynomials.
Averages of products of matrix elements in the circular ensembles can be calculated using Weingarten functions.
For large dimension of the matrix these calculations become impractical, and a numerical method is advantageous.
There exist efficient algorithms to generate random matrices in the circular ensembles, for example by performing a QR decomposition on a Ginibre matrix.