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. 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 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: λ k = e i θ k {displaystyle lambda _{k}=e^{i heta _{k}}} with 0 ≤ θ k < 2 π {displaystyle 0leq heta _{k}<2pi } for k=1,2,... n, where the θ k {displaystyle heta _{k}} are also known as eigenangles or eigenphases. In the CSE each of these n eigenvalues appears twice. The distributions have densities with respect to the eigenangles, given by on R [ 0 , 2 π ] n {displaystyle mathbb {R} _{}^{n}} (symmetrized version), where β=1 for COE, β=2 for CUE, and β=4 for CSE. 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 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 e i θ k {displaystyle e^{i heta _{k}}} and e − i θ k {displaystyle e^{-i heta _{k}}} , 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 θk 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