512
G Circular Ensembles
where O
1
N
indicates that we have neglected terms of order
1
N . Equation (G.114)
gives the 3 -statistic for the circular orthogonal ensemble.
G.6.3 Circular Symplectic Ensemble
The two-body cluster function for the circular symplectic ensemble can be obtained
from the matrix equation (see Eq. (6.48))
T N 4 (θ 1 , θ 2 )
0
0
T N 4 (θ 1 , θ 2 )
= − ¯
σ N,4 (θ 1 , θ 2 )· ¯
σ N,4 (θ 1 , θ 2 ).
(G.115)
From Eq. (G.100), we obtain
T N 4 (θ 1 , θ 2 ) = −
1
4
[S 2N (θ 1 − θ 2 )S 2N (θ 2 − θ 1 ) − DS 2N (θ 1 − θ 2 )I S 2N (θ 1 − θ 2 )],
(G.116)
where
S 2N (θ 1 − θ 2 ) =
1
2π
sin[N(θ 1 − θ 2 )]
sin[(θ 1 − θ 2 )/2]
=
2
D
sin[2πχ]
2πχ
,
(G.117)
with χ = ξ 2 − ξ 1 = (θ 1 − θ 2 )/D,
DS N (θ 1 − θ 2 ) = −
d
dθ 2
S N (θ 1 − θ 2 ) =
2
D
d
dχ
sin[2πχ]
2πχ
,
(G.118)
and
I S N (θ 1 − θ 2 ) =
θ 1 −θ 2
0
dθ
S N (θ
) =
2
D
χ
0
ds
sin[2πs]
2πs
.
(G.119)
If we combine Eqs. (G.116)–(G.119), the two-body cluster function takes the
form
T N 4 (θ 1 , θ 2 ) =
1
D 2 Y
(CSE)
4
(ξ 1 , ξ 2 ),
(G.120)
where Y
(CSE)
4
(ξ 1 , ξ 2 ) = Y
(GSE)
4
(ξ 1 , ξ 2 ) (see Eq. (G.120)) with χ = ξ 1 − ξ 2 . The
circular symplectic ensemble and the Gaussian symplectic ensemble have the same
3 -statistic, namely
(CSE)
3
=
1
4π 2
ln[4πN] + γ −
5
4
+
π 2
8
+ O
1
N
. . .,
(G.121)