482
F Gaussian Ensembles
F.2.5 Eigenvalue Number Density for GUE
We can derive exact expressions for the eigenvalue number density for the Gaussian
unitary ensemble. The eigenvalue number density is given by
ρ
c
N (x) = T
C
N (x).
(F.83)
The average number of eigenvalues in the energy interval x→x + dx is ρ N 2 (x)dx,
where
∞
−∞
ρ
c
N (x)dx = N.
(F.84)
We now can write
ρ
c
N (x) = T
C
N (x) = ¯
σ
c
N (x, x).
(F.85)
The eigenvalue density for the Gaussian unitary ensemble (GUE) takes the form
ρ
c
N (x) =
N −1
k=0
φ k (x)φ k (x).
(F.86)
A plot the eigenvalue number density, ρ c
N (x), for the case N = 22 is gives a result
very similar to the plot of the GOE number density (for N = 22) given in Fig. 6.1a.
F.2.6 3 -Statistics for GUE
The 3 -statistic is completely determined by the form of the two-body cluster function, T C
N (x 1 , x 2 ).The two-body cluster function for the Gaussian unitary ensemble
can be written
T
C
N (x 1 , x 2 ) = − ¯
σ
c
N,2 (x 1 , x 2 )· ¯
σ
c
N,2 (x 2 , x 1 ) = −
N −1
n=0
φ n (x 1 )φ n (x 2 )
2
.
(F.87)
The summation in Eq. (F.87) can be written
N −1
n=0
φ n (x 1 )φ n (x 2 ) =
N
2
φ N (x 1 )φ N −1 (x 2 ) − φ N (x 2 )φ N −1 (x 1 )
x 1 − x 2
.
(F.88)
Equation (F.88) is called the Christoffel-Darboux formula (Bateman 1953; Mehta
1991).