G.4 Circular Symplectic Ensemble (COE)
507
The factor
1
(4π) N in Eq. (G.84) comes from the coefficients in Eqs. (G.76)
and (G.77).
Let us now note that
Det[ ¯
M
D
N ] = Det[ ¯
Y N ] and (−i)
2N
N −
1
2
n=−N +
1
2
n
Det[ ¯
M
I
N ] = Det[ ¯
Y N ],
(G.88)
where
N −
1
2
n=−N +
1
2
n =
(−1) N (2N) 2
(2 2N −1 ) 2 (N) 2 .
(G.89)
The determinant of ¯
Q
(4)
N becomes
Det[ ¯
Q
(4)
N ] =
1
(4π) 2N
(2 2N −1 ) 2 [N] 2
(2N) 2
|Det[ ¯
Y N ]|
2 .
(G.90)
The quaternion ¯
Q
(4)
N is self-dual, so we can write the quaternion determinant of
¯
Q
(4)
N as
Qdet[ ¯
Q
(4)
N ] =
1
(4π) N
2 2N −1 (N)
(2N)
|Det[ ¯
Y N ]|
1 0
0 1
.
(G.91)
Since both the probability density, P N 4 (θ 1 , . . ., θ N ), obtained in Eq. (G.74) and
Qdet[ ¯
Q
(4)
N ] are proportional to |Det[ ¯
Y N ]|, the probability density for the circular
symplectic ensemble can be written
P N 4 (θ 1 , . . ., θ N ) ¯
1 2 =
1
N!
Qdet[ ¯
Q
(4)
N ].
(G.92)
Therefore,
P N 4 (θ 1 , . . ., θ N ) =
1
N! (4π) N
2 2N −1 (N)
(2N)
|Det[ ¯
Y N ]|
= C N,4
1≤j |e
iθ j − e
iθ i |
4 .
(G.93)
Since |Det[ ¯
Y N ]| =
1≤j C N,4 =
1
N!
1
(4π) N
2 2N −1 (N)
(2N)
=
1
π N (2N)!
.
(G.94)