6.4 Eigenvalue Statistics: Gaussian Orthogonal Ensemble
177
Let us now consider the first term in Eq. (6.95),
K N (x 1 , x 2 ) ≡
N
2 −1
n=0
φ 2n (x 1 )φ 2n (x 2 ) =
1
2
N −1
n=0
[φ n (x 1 )φ n (x 2 ) + φ n (−x 1 )φ n (x 2 )].
(6.96)
If we take the limits N→∞, x 1 →0, and x 2 →0 so that ξ = x 1 /D and η = x 2 /D
remain finite, we obtain
lim
N →∞
lim
x 1 ,x 2 →0
K N (x 1 , x 2 ) =
1
D
Q(ξ, η),
(6.97)
where
Q 1 (ξ, η) =
1
2
sin[π(ξ − η)]
π(ξ − η)
+
sin[π(ξ + η)]
π(ξ + η)
.
(6.98)
Thus, in the limits N →∞, x 1 →0, and x 2 →0 so that ξ = x 1 /D and η = x 2 /D
remain finite, we can write
S
N
1,1 (x 1 , x 2 ) =
1
D
Q 1 (ξ, η) −
d
dξ
η
0
ds Q 1 (ξ, s)
=
1
D
sin[πχ]
πχ
,
(6.99)
where χ = ξ − η. Similarly, we can write
S
N
1,2 (x 1 , x 2 ) =
1
D 2
d
dχ
sin[πχ]
πχ
(6.100)
and
I S
N
2,1 (x 1 , x 2 ) =
∞
−∞
dt t(x 1 − t) S
N
1,1 (t, x 2 ) =
ξ −η
0
ds
sin[πs]
πs , if ξ > η
−
η−ξ
0
ds
sin[πs]
πs , if η > ξ.
(6.101)
If we combine the above equations, we obtain the expression for the two-body
cluster function
T N (x 1 , x 2 ) = −S
N
1,1 (x 1 , x 2 )
2
+ S
N
1,2 (x 1 , x 2 )S
N
2,1 (x 1 , x 2 )
=
1
D 2 Y
(GOE)
2
(ξ, η),
(6.102)
where
Y
(GOE)
2
(ξ, η) = −
sin[πχ]
πχ
2
+
1
2
−
χ
0
ds
sin[πs]
πs
d
dχ
sin[πχ]
πχ
,
(6.103)
with χ = ξ − η.