510
G Circular Ensembles
We can now use the results of this section to compute various measurable properties
of the spectrum of Hermitian and unitary matrices.
G.6 3 -Statistics for Circular Ensembles
In this section, we obtain expressions for the 3 -statistics for the three circular
ensembles. We assume that we are given a sequence of N eigenvalues, {θ j }, and
that they have an average spacing D = 2π/N. We will rescale the eigenvalues
{θ j } so they have unit average spacing. This can be done by setting θ j = Dξ j =
2πξ j /N. Since the rescaled eigenvalues, {ξ j }, have unit average spacing, we can
use the formalism developed in Sect. 6.4.2.
Below we obtain the 3 -statistics for the circular unitary ensemble, the circular
orthogonal ensemble, and the circular symplectic ensemble. We will find that they
are the same as their counterparts in the Gaussian ensembles.
G.6.1 Circular Unitary Ensemble
The two-body cluster function for the circular unitary ensemble can be written (see
Eq. (6.48))
T N 2 (θ 1 , θ 2 ) = − ¯
σ N,2 (θ 1 , θ 2 )· ¯
σ N,2 (θ 2 , θ 1 ) =
1
2π
sin[N(θ i − θ j )/2]
sin[(θ i − θ j )/2]
2
.
(G.104)
Let us rescale the variables so that θ j = Dξ j = 2πξ j /N. Then, in the limit N→∞,
the two-body cluster function, T N 2 (θ 1 , θ 2 ), takes the form
T N 2 (x 1 , x 2 ) =
1
D 2 Y
(CU E)
2
(ξ 1 , ξ 2 ),
(G.105)
where Y
(CU E)
2
(ξ 1 , ξ 2 ) = Y
(GU E)
2
(ξ 1 , ξ 2 ) (see Eq. (6.85)). Since the two unitary
ensembles have the same two-body cluster function, Y 2 (ξ 1 , ξ 2 ), they will have the
same 3 -statistics. Therefore,
(CU E)
3
=
1
2π 2
ln[2πN] + γ −
5
4
+ O
1
N
. . .,
(G.106)
where O
1
N
indicates that we have neglected terms of order
1
N . Equation (G.106)
gives the 3 -statistic for the circular unitary ensemble.
G Circular Ensembles
We can now use the results of this section to compute various measurable properties
of the spectrum of Hermitian and unitary matrices.
G.6 3 -Statistics for Circular Ensembles
In this section, we obtain expressions for the 3 -statistics for the three circular
ensembles. We assume that we are given a sequence of N eigenvalues, {θ j }, and
that they have an average spacing D = 2π/N. We will rescale the eigenvalues
{θ j } so they have unit average spacing. This can be done by setting θ j = Dξ j =
2πξ j /N. Since the rescaled eigenvalues, {ξ j }, have unit average spacing, we can
use the formalism developed in Sect. 6.4.2.
Below we obtain the 3 -statistics for the circular unitary ensemble, the circular
orthogonal ensemble, and the circular symplectic ensemble. We will find that they
are the same as their counterparts in the Gaussian ensembles.
G.6.1 Circular Unitary Ensemble
The two-body cluster function for the circular unitary ensemble can be written (see
Eq. (6.48))
T N 2 (θ 1 , θ 2 ) = − ¯
σ N,2 (θ 1 , θ 2 )· ¯
σ N,2 (θ 2 , θ 1 ) =
1
2π
sin[N(θ i − θ j )/2]
sin[(θ i − θ j )/2]
2
.
(G.104)
Let us rescale the variables so that θ j = Dξ j = 2πξ j /N. Then, in the limit N→∞,
the two-body cluster function, T N 2 (θ 1 , θ 2 ), takes the form
T N 2 (x 1 , x 2 ) =
1
D 2 Y
(CU E)
2
(ξ 1 , ξ 2 ),
(G.105)
where Y
(CU E)
2
(ξ 1 , ξ 2 ) = Y
(GU E)
2
(ξ 1 , ξ 2 ) (see Eq. (6.85)). Since the two unitary
ensembles have the same two-body cluster function, Y 2 (ξ 1 , ξ 2 ), they will have the
same 3 -statistics. Therefore,
(CU E)
3
=
1
2π 2
ln[2πN] + γ −
5
4
+ O
1
N
. . .,
(G.106)
where O
1
N
indicates that we have neglected terms of order
1
N . Equation (G.106)
gives the 3 -statistic for the circular unitary ensemble.