F.1 Gaussian Orthogonal Ensemble (GOE)
471
where
Det[ ¯
K N ] = Det
φ 0 (1)
φ 0 (2) ... φ 0 (N)
φ
0 (1)
φ
0 (2) . . . φ
0 (N)
φ 2 (1)
φ
2 (2) . . . φ 2 (N)
φ
2 (1)
φ
2 (2) . . . φ
2 (N)
. . .
. . .
. . .
. . .
φ N −2 (1) φ N −2 (2) . . . φ N −2 (N)
φ
N −2 (1) φ
N −2 (2) . . . φ
N −2 (N)
.
(F.13)
If we now use Eqs. (F.9) and (F.12), we obtain (after some algebra)
P N 1 (x 1 , . . ., x N ) = C N,1
2 N
N!
N
k=1
1 +
k
2
|Det[ ¯
K N ]|.
(F.14)
In Eq. (F.14), we have obtained an expression for P N 1 (x 1 , . . ., x N ) in terms of
|Det[ ¯
K N ]|. This will prove useful below, where we construct an expression for
P N 1 (x 1 , . . ., x N ) in terms of an N×N quaternion. From Eq. (6.33) we can determine
the normalization constant C N , and find that the probability density takes the form
P N 1 (x 1 , . . ., x N ) =
1
N!
1
2 N/2 |Det[ ¯
K N ]|.
(F.15)
In Appendix E, we show that the quaternion determinant (denoted Qdet) of an
N×N quaternion matrix ¯
Q N has the following relation to the determinant of the
matrix ¯
K N ,
Qdet[ ¯
Q N ] =
1
2 N/2 |Det[ ¯
K N ]|
1 0
0 1
.
(F.16)
Therefore, the probability density P N 1 (x 1 , . . ., x N ) can be written in terms of the
quaternion determinant of the N×N quaternion matrix ¯
Q N as
P N (x 1 , . . ., x N ) ¯
1 =
1
N!
Qdet[ ¯
Q N ],
(F.17)
where ¯
1 is the 2×2 unit matrix, and the quaternion matrix ¯
Q N is an N×N matrix
whose matrix elements are 2×2 matrices
¯
Q N =
⎛
⎜
⎜
⎜
⎝
¯
σ N (1, 1) ¯
σ N (1, 2) . . . ¯
σ N (1, N)
¯
σ N (2, 1) ¯
σ N (2, 2) . . . ¯
σ N (2, N)
. . .
. . .
. . .
. . .
¯
σ N (N, 1) ¯
σ N (N, 2) . . . ¯
σ N (N, N)
⎞
⎟
⎟
⎟
⎠
.
(F.18)