G.5 Cluster Functions for Circular Ensembles
509
where p ranges over odd half-integers, (p = −
N −1
2 , −
N −3
2 , . . ., +
N −1
2 ). The
remaining matrix elements are given by
DS N (θ ) =
d
dθ
S N (θ ) =
1
2π
+
N
2 −
1
2
p=−
N
2 +
1
2
ipe
ipθ
(G.98)
and
JS N (θ ) = −
1
π
∞
k=0
sin[(N + 2k + 1) θ/2]
[(N + 2k + 1)/2]
.
(G.99)
G.5.3 Circular Symplectic Ensemble (CSE)
The expression for the 2×2 matrix, ¯
σ N 4 (θ i , θ j ), for the Gaussian symplectic
ensemble (see Eq. (G.76)) is
¯
σ N 4 (θ i , θ j ) =
1
2
S 2N (θ i − θ j ) DS 2N (θ i − θ j )
IS 2N (θ i − θ j ) S 2N (θ i − θ j )
.
(G.100)
The matrix element, S 2N (θ i − θ j ), is defined as
S 2N (θ ) =
1
2π
N −
1
2
p=−N +
1
2
e
ipθ
=
1
2π
sin[Nθ]
sin[θ/2]
,
(G.101)
where p ranges over all half-integer values p = −N +
1
2 , −N +
3
2 , . . ., N −
1
2 .
Similarly,
DS 2N (θ ) =
d
dθ
S 2N (θ ) =
N −
1
2
p=−N +
1
2
ip e
ipθ ,
(G.102)
and
IS 2N (θ ) =
θ
0
dθ
S 2N (θ
) =
N −
1
2
p=−N +
1
2
1
ip
e
ipθ .
(G.103)