8.6 COE and GOE
277
P
C ( ¯
U)dd U =
η N
U
δδ O
π
−π
dθ 1 . . .
π
−π
dθ N
1≤i |e
iθ j − e
iθ i | = 1.
(8.121)
If we integrate over the invariant measure δδ O , we obtain
P
C ( ¯
U)dd U = C N
π
−π
dθ 1 . . .
π
−π
dθ N
1≤i |e
iθ j − e
iθ i |
β
= 1,
(8.122)
where
C N =
η N δO
U
.
(8.123)
The joint probability density to find the phase angles in the intervals θ j →θ j + dθ j
for (j = 1, 2, . . . , N) takes the form
P
C
N (θ 1 , . . . , θ N )dθ 1 . . . dθ N = C N
1≤i |e
iθ j − e
iθ i |
β dθ 1 . . . dθ N .
(8.124)
The normalization constant, C N , can be obtained by requiring that
π
−π
dθ 1 . . .
π
−π
dθ N P N β (θ 1 , . . . , θ N ) = 1
(8.125)
and it is given by
C N =
1
(2π) N
(
3
2 ) N
(1 +
1
2 N)
(8.126)
(Dyson 1962a; Wilson 1962).
8.6.2 Lorentzian Orthogonal Ensembles
The Lorentzian orthogonal ensembles are ensembles of real symmetric random
N ×N Hamiltonian matrices, ¯
H , whose matrix elements are determined by a
probability density, P
N ( ¯
H ), of the form
P
N ( ¯
H )dd H = ˜
C N
λ N(N+1)/2 dd H
Det[λ 2 ¯
1 N + ( ¯
H − −e ¯
1) 2 ] (N +1)/2
,
(8.127)
where C N is the normalization constant, e is the center of the probability
distribution, and λ is the width of the distribution. The case when the Hamiltonian is