6.4 Eigenvalue Statistics: Gaussian Orthogonal Ensemble
167
where
i , x j ) = i − x j ) −
1
2
(6.56)
( i − x j ) is the Heaviside function and has the property that = 0 for x < 0,
=
1
2 for x = 0, and = 1 for x > 0). The functions φ n (x) are harmonic
oscillator wave functions
φ n (x) = (
√
π 2
n n!)
−
1
2 exp
−
1
2
x
2
H n (x),
(6.57)
where H n (x) are Hermite polynomials. The harmonic oscillator wave functions are
orthonormal in the sense that
∞
−∞
dx φ i (x)φ j (x) =
1, if i = j ;
0, if i =j.
(6.58)
We can now use these results to explore predictions of the Gaussian Orthogonal
Ensemble.
6.4 Eigenvalue Statistics: Gaussian Orthogonal Ensemble
We now consider various statistical measures of the eigenvalue statistics in the
Gaussian Orthogonal Ensemble. We begin with the eigenvalue number density. We
then consider a measure of the correlation between eigenvalues (the 3 -statistic).
Finally, we consider the eigenvalue nearest neighbor spacing distribution.
6.4.1 Eigenvalue Number Density
Given the expressions for the cluster functions in Eq. (6.49), it is straightforward
to derive expressions for eigenvalue number density for the Gaussian Orthogonal
Ensemble.
Let us consider a system with N eigenvalues, {x i }, distributed with a probability
density P (x 1 , . . . , x N ). The eigenvalue number density operator is
ˆ
ρ(x) =
N
i=1
δ(x − x i ),
(6.59)
where the “hat” on ˆ
ρ(x) indicates that it depends on the eigenvalues x 1 , . . . , x n .
The eigenvalue number density, ρ N (x), is the average of ˆ
ρ(x) with respect to the
distribution P N (x 1 , . . . , x N ),