494
G Circular Ensembles
the circular symplectic ensemble we set β = 4. The volume, ¯
U , is given in
Appendix D for β = 1 and β = 2.
It is useful to write the probability density, P ( ¯
U), in polar form. From Eq. (D.80),
we can write
P( ¯
U)dd U =
η N,β
U
δδ V ,β
π
−π
dθ 1 . . .
π
−π
dθ N
1≤i |e
iθ j − e
iθ i |
β
= 1.
(G.4)
If we integrate over the invariant measure δδ V ,β , we obtain
P( ¯
U)dd U = C N (β)
π
−π
dθ 1 . . .
π
−π
dθ N
1≤i |e
iθ j − e
iθ i |
β
= 1,
(G.5)
where
C N (β) =
η N,β δV ,β
U
.
(G.6)
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 N β (θ 1 , . . ., θ N )dθ 1 . . .dθ N = C N (β)
1≤i |e
iθ j − e
iθ i |
β dθ 1 . . .dθ N .
(G.7)
The normalization constant, C N (β), can be obtained by requiring that
π
−π
dθ 1 . . .
π
−π
dθ N P N β (θ 1 , . . ., θ N ) = 1
( G . 8 )
and it is given by
C N (β) =
1
(2π) N
(1 +
1
2 β) N
(1 +
1
2 Nβ)
(G.9)
(see Appendix D and (Dyson 1962a; Wilson 1962).
In this appendix, we derive a unified formulation of the three circular ensembles
and determine the normalization constants, C N,β , for the probability densities
P N β (θ 1 , . . ., θ N ) = C N,β
1≤i |e
θ j − e
θ i |
β .
(G.10)
The probability densities satisfy the normalization condition,
π
−π
dθ 1 . . .
π
−π
dθ N P N β (θ 1 , . . ., θ N ) = 1.
(G.11)