Circular Ensemble - Generalizations

Generalizations

Generalizations of the circular ensemble restrict the matrix elements of U to real numbers 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 and, possibly complemented by eigenvalues fixed at +1 or -1. For n=2m even and det U=1, there are no fixed eigenvalues and the phases θ_k have probability distribution

with C an unspecified normalization constant. For n=2m+1 odd there is one fixed eigenvalue σ=det U equal to ±1. The phases have distribution

For n=2m+2 even and det U=-1 there is a pair of eigenvalues fixed at +1 and -1, while the phases have distribution

This is also the distribution of the eigenvalues of a matrix in Sp(2m).

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.