508
G Circular Ensembles
This expression for C N,4 agrees with Eq. (G.13). Thus, we have been able to express
the probability density for the circular symplectic ensemble (CSE) in the general
form shown in Eq. (G.12), and we have obtained the normalization constant.
G.5 Cluster Functions for Circular Ensembles
In this subsection, we give expressions for ¯
σ N β (θ i , θ j ) for the circular unitary
ensemble (CUE), the circular orthogonal ensemble (COE), and the circular symplectic ensemble (CSE). The derivation of ¯
σ N β (θ i , θ j ) for all three ensembles was
obtained in the previous sections. The expressions for ¯
σ N β (θ i , θ j ), together with
Eqs. (6.47)–6.49), allow us to write the cluster functions for the three circular
ensembles.
G.5.1 Circular Unitary Ensemble (CUE)
The expression for the scalar function, ¯
σ N 2 (θ i , θ j ), for the Gaussian unitary
ensemble was obtained in Eq. (G.21). It is
σ N 2 (θ i , θ j ) =
1
2π
N
k=1
e
i(2k−1−N )(θ i −θ j )/2
=
1
2π
sin[N(θ i − θ j )/2]
sin[(θ i − θ j )/2]
.
(G.95)
G.5.2 Circular Orthogonal Ensemble (COE)
We will only consider the case for N = 2m, where m is a positive integer. (The
case for N odd can be found in Mehta 1991.) The expression for the 2×2 matrix,
¯
σ N 1 (θ i , θ j ), for N even (see Eq. (G.31)) is
¯
σ N 1 (θ i , θ j ) =
S N (θ i − θ j ) DS N (θ i − θ j )
JS N (θ i − θ j ) S N (θ j − θ i )
.
(G.96)
The matrix element, S N (θ ), in Eq. (G.96) is given by
S N (θ ) =
1
2π
+
N
2 −
1
2
p=−
N
2 +
1
2
e
ipθ
=
sin[
N
2 θ ]
2π sin[
1
2 θ ]
,
(G.97)