F.2 Gaussian Unitary Ensemble (GUE)
481
and
∞
−∞
dy σ
c
N (x i , y)σ
c
N (y, x j ) = σ
c
N (x i , x j ).
(F.75)
Using Eqs. (F.74) and (F.75), we find that
∞
−∞
dx n+1 . . .
∞
−∞
dx N Det[ ¯
Q
C
N ] = (N − n)! Det[ ¯
Q
C
n (N)],
(F.76)
where ¯
Q C
n (N) is an n×n quaternion matrix composed of matrix elements
σ c
N (x i , x j ), and
∞
−∞
dx 1 . . .
∞
−∞
dx N Det[ ¯
Q
C
N ] = N!.
(F.77)
If we now combine the above equations, we obtain
P C (x 1 , . . ., x N ) =
1
N!
Det[ ¯
Q
C
N ],
(F.78)
which is the probability density for the Gaussian unitary ensemble (GUE) written in
terms of an N ×N quaternion matrix.
The reduced n-level probability density, R C
N (x 1 , . . ., x n ) is defined
R
C
N (x 1 , . . ., x n ) =
N!
(N − n)!
∞
−∞
. . .
∞
−∞
dx n+1 . . .dx N P C (x 1 , x 2 , . . ., x N ).
(F.79)
From Eq. (F.26), we can write this in the form
R
C
N β
(x 1 , . . ., x n ) ¯
1 = Qdet[ ¯
Q
C
n ].
(F.80)
From Appendix E, we know that the quaternion determinants, Qdet[ ¯
Q C
n ], naturally
decompose into clusters of connected products of matrix elements, ¯
σ c
N (x i , x j ).
Thus, we find that the n-level reduced probability densities decompose into fully
connected m-level cluster functions T C
N (x 1 , . . ., x m ) so that
R
C
N (x 1 ) = T
C
N (x 1 ),
(F.81)
R
C
N (x 1 , x 2 ) = T
C
N (x 1 , x 2 ) + T
C
N (x 1 )T
C
N (x 2 ),
(F.82)
etc.