F.2 Gaussian Unitary Ensemble (GUE)
483
Let us rescale the variables x 1 and x 2 , and let x 1 = Dξ and x 2 = Dη, where
D = π/
√
2N. For the case of N even, we can write (Mehta 1991)
lim
N →∞
(−1)
N/2
N
2
1/4
φ N (x) =
1
√
π
cos(π ξ ),
lim
N →∞
(−1)
N/2
N
2
1/4
φ N −1 (x) = −
1
√
π
sin(π ξ ).
(F.89)
Thus, as N→∞, and x 1 →0 and x 2 →0 (keeping ξ and η finite), we obtain
lim
N →∞
lim
x 1 ,x 2 →0
D
N −1
k=0
φ k (x 1 )φ k (x 2 ) =
sin[π(ξ − η)]
π(ξ − η)
,
(F.90)
and finally the two-body cluster function takes the form
Y
(GU E)
2
(ξ 1 , ξ 2 ) = D
2 T N 2 (x 1 , x 2 ) = −
sin[π(ξ 2 − ξ 1 )]
π(ξ 2 − ξ 1 )
2
.
(F.91)
We will be interested in the limit of large N. It is useful to note that
N
0
dx
sin(π x) 2
(π x) 2 =
1
2
−
1
2Nπ 2 + . . .
(F.92)
and
N
0
dx x
sin(π x) 2
(π x) 2 =
1
2π 2
γ + ln[2Nπ] −
1
(2Nπ) 2
+ . . .,
(F.93)
where γ = 0.577216 is the Euler gamma constant. If we now perform the
integrations, we obtain
ˆ
η
2
eig =
4
3
m
2
+ +
1
π 2 [γ + ln[4π m]] + . . .,
(F.94)
ˆ
η
2
eig = m
2
+
1
4π 2 +
1
2π 2 [γ + ln[4πm]] + . . .,
(F.95)
and
ξ ˆ
η
2
eig =
m 4
9
+
m 2
8π 2 .
(F.96)
We can combine Eqs. (F.94), and (F.96) to obtain