G.6 3 -Statistics for Circular Ensembles
511
G.6.2 Circular Orthogonal Ensemble
The matrix equation that gives us the two-body cluster function for the circular
orthogonal ensemble is (see Eq. (6.48))
T N 1 (θ 1 , θ 2 )
0
0
T N 1 (θ 1 , θ 2 )
= − ¯
σ N,1 (θ 1 , θ 2 )· ¯
σ N,1 (θ 2 , θ 1 ).
(G.107)
Therefore, we obtain
T N 1 (θ 1 , θ 2 ) = −S N (θ 1 −θ 2 )S N (θ 2 −θ 1 )−DS N (θ 1 −θ 2 )J S N (θ 2 −θ 1 ),
(G.108)
where
S N (θ 1 − θ 2 ) =
1
2π
sin[N(θ i − θ j )/2]
sin[(θ i − θ j )/2]
=
1
D
sin[πχ]
πχ
,
(G.109)
with χ = ξ 2 − ξ 1 ,
DS N (θ 1 − θ 2 ) = −
d
dθ 2
S N (θ 1 − θ 2 ) =
1
D 2
d
dχ
sin[πχ]
πχ
,
(G.110)
J S N (θ 1 − θ 2 ) = I S N (θ 1 − θ 2 ) − (θ 1 − θ 2 )
=
θ 1 −θ 2
0
dθ
S N (θ
) − (θ 1 − θ 2 ) =
χ
0
ds
sin[πs]
πs
− (χ), (G.111)
and
1 − θ 2 ) = (χ) =
+
1
2 , if 0 < χ≤π
−
1
2 , if − π ≤χ < 0.
(G.112)
If we now combine Eqs. (G.108)–(G.112), we obtain an expression for the twobody cluster function,
T N 1 (θ 1 , θ 2 ) =
1
D 2 Y
(COE)
2
(ξ 1 , ξ 2 ),
(G.113)
where Y
(COE)
2
(ξ 1 , ξ 2 ) = Y
(GOE)
2
(ξ 1 , ξ 2 ) for χ > 0, which is the case of interest
when computing the 3 -statistic. Thus, the two ensembles have the same 3 -
statistic, namely
(COE)
3
=
1
π 2
ln[2πN] + γ −
5
4
−
π 2
8
+ O
1
N
. . .,
(G.114)