504
G Circular Ensembles
G.4 Circular Symplectic Ensemble (COE)
The probability density for the circular symplectic ensemble is given by
P N 4 (θ 1 , . . ., θ N ) = C N,4
1≤i |e
iθ j − e
iθ i |
4 ,
(G.68)
where C N,4 is a normalization constant. We can write the product of phase
differences in the form
1≤i (e
iθ j − e
iθ i )
4
= Det[ ¯
W N ],
(G.69)
where W N is an N×N quaternion matrix whose (m, n)th matrix element is
( ¯
W N ) m,n =
e i(2m−2)θ n (2m − 2)e i(2m−3)θ n
e i(2m−1)θ n (2m − 1)e i(2m−2)θ n
.
(G.70)
For example, ¯
W 2 is given by
¯
W 2 =
⎛
⎜
⎜
⎝
1
0
1
0
e iθ 1
1 e iθ 2
1
e i2θ 1 2e iθ 1 e i2θ 2 2e iθ 2
e i3θ 1 3e i2θ 1 e i3θ 2 3e i2θ 2
⎞
⎟
⎟
⎠ .
(G.71)
If we use Eq. (G.15), we obtain
1≤j |e
iθ k − e
iθ j |
4
= e
−2i(N−1)
N
j =1 θ j Det[ ¯
W N ] = Det[ ¯
Y N ],
(G.72)
where ¯
Y N is an N ×N quaternion matrix whose (i, j )th matrix element is the 2×2
matrix
( ¯
Y N ) i,j =
e
−
i
2 (2N −(2j −1))θ i −i
d
dθ i
e
−
i
2 (2N −(2j −1))θ i
e
−
i
2 (2N −(2j +1))θ i −i
d
dθ i
e
−
i
2 (2N −(2j +1))θ i
.
(G.73)
To obtain Eqs. (G.72) and (G.73), we have multiplied each of the columns in
Det[ ¯
W N ] that depend on θ j by a factor e
−
i
2 (2N −1)θ j . We have also added multiples
of odd-numbered columns to even-numbered columns (this does not change the
value of the determinant). Thus, we find that
P(θ 1 , . . ., θ N ) = C N,4 Det[ ¯
Y N ],
(G.74)