502
G Circular Ensembles
where ¯
0 N is an N ×N matrix with every matrix element equal to zero,
¯
S N =
⎛
⎜
⎜
⎜
⎝
S N (θ 1 − θ 1 ) S N (θ 1 − θ 2 ) . . . S N (θ 1 − θ N )
S N (θ 2 − θ 1 ) S N (θ 2 − θ 2 ) . . . S N (θ 2 − θ N )
. . .
. . .
. . .
. . .
S N (θ N − θ 1 ) S N (θ N − θ 2 ) . . . S N (θ N − θ N )
⎞
⎟
⎟
⎟
⎠
,
(G.56)
¯
S N =
⎛
⎜
⎜
⎜
⎝
0
D S N (θ 1 − θ 2 ) . . . DS N (θ 1 − θ N )
DS N (θ 2 − θ 1 )
0
. . . DS N (θ 2 − θ N )
. . .
. . .
. . .
. . .
DS N (θ N − θ 1 ) DS N (θ N − θ 2 ) . . .
0
⎞
⎟
⎟
⎟
⎠
(G.57)
and
¯
N =
⎛
⎜
⎜
⎜
⎝
0
− N (θ 1 − θ 2 ) . . . − N (θ 1 − θ N )
− N (θ 2 − θ 1 )
0
. . . − N (θ 2 − θ N )
. . .
. . .
. . .
. . .
− N (θ N − θ 1 ) − N (θ N − θ 2 ) . . .
0
⎞
⎟
⎟
⎟
⎠
.
(G.58)
We now can rearrange Det[ ¯
Q
(1)
N ] in a manner similar to that described in
Appendix F, and we find
Det[ ¯
Q
(1)
N ] = Det[¯ N ]×Det[ ¯
S N ].
(G.59)
Let us next note that ¯
S N is a product of two N ×N matrices,
¯
S N =
⎛
⎜
⎜
⎜
⎜
⎝
¯
†
1
¯
†
2
. . .
¯
†
N
⎞
⎟
⎟
⎟
⎟
⎠
·
−¯ p 1 −¯ p 2 . . . −¯ p N
.
(G.60)
Note also that Det[¯ N ] =
1
2 N . Thus we obtain
Det[ ¯
Q
(1)
N ] =
1
2 N
1
(2π) N (i)
N
N
2 −
1
2
p=−
N
2 +
1
2
p
|Det[ ¯
U N ]|
2 ,
(G.61)
where ¯
U N is defined in Eq. (G.17). But