490
F Gaussian Ensembles
If we now use Eq. (F.121), we obtain
S 2N +1 (x 1
√
2, x 2
√
2) =
√
2
D
Q 4 (ξ, η) −
d
dξ
η
−∞
ds Q 4 (ξ, s)
=
√
2
D
sin[2π(χ)]
2π(χ)
,
(F.131)
where χ = ξ − η. Similarly, we can write
DS 2N +1 (x 1
√
2, x 2
√
2) =
1
D 2
d
dχ
sin[2π(χ)]
2π(χ)
,
(F.132)
and
I S 2N +1 (x 1
√
2, x 2
√
2) =
√
2
∞
−∞
dt t(x 1 − t) S 2N +1 (x 1
√
2, x 2
√
2)
=
2
ξ −η
0
ds
sin[πs]
πs , ifξ > η
−2
η−ξ
0
ds
sin[πs]
πs , ifη > ξ.
(F.133)
If we combine Eqs. (F.129) and (F.131)–(F.133), we obtain the expression for the
two-body cluster function
T N 4 (x 1 , x 2 ) =
1
D 2 Y
(GSE)
4
(ξ, η),
(F.134)
where
Y
(GSE)
4
(ξ, η) = −
sin[2πχ]
(2πχ)
2
+
d
dχ
sin[2πχ]
2πχ
χ
0
ds
sin[2πs]
2πs
,
(F.135)
with χ = ξ − η.
Given the form of the cluster function, Y
(GSE)
4
(ξ, η), in Eq. (F.135), it is
straightforward to compute the 3 -statistic. From the definition of the 3 -statistic
given in Eqs. (6.80)–(6.88), we obtain
(GSE)
3
=
1
4π 2
ln[4πN] + γ −
5
4
+
π 2
8
+ O
1
N
. . .,
(F.136)
where O
1
N
indicates that we have neglected terms of order
1
N . Equation (F.136)
gives the 3 -statistic for the Gaussian symplectic ensemble. A plot of
(GSE)
3
would
give a curve (not shown) that lies slightly below the curve for GUE in Fig. 6.6.