The Gaussian Unitary Ensemble (GUE) is the set of all hermitian
matrices together with a unique probability measure Pb(H)Õi=1N dxi with the following properties:
If H is a hermitian matrix and U is a unitary matrix then Pb (H) = Pb (UHU*)
If H has a side of length N then Pb (H) is a product of N2 functions,
each of which depends on a single variable.
Property (1) states that Pb (H) depends only on the
eigenvalues of H. To see this, recall that any hermitian matrix
is diagonalizable via conjugation by some unitary matrix. If U is
a unitary matrix that diagonalizes H and if D = UHU* then the
diagonal elements of D are exactly the eigenvalues of H in some
order and Pb (D) = Pb (H). Property (2) says that
Pb makes the elements of H into independent random
variables. With some work it follows from these properties that
for the GUE of N×N matrices,
PNb (H)
N Õ
i=1
dxi = (ZNb)-1exp(
-Nb
2
N å
i=1
xi2)
Õ
1 £ i < j £ N
|xi - xj|b
N Õ
i=1
dxi
where
ZNb =
ó õ
RN
exp(
-Nb
2
N å
i=1
xi2)
Õ
1 £ i < j £ N
|xi - xj|b
N Õ
i=1
dxi.
We want to
consider Hermitian matrices with a probability measure PNb(H)Õi=1N dxi as above, but we replace [(x2)/2]
with some function V in the exponential, where V diverges nicely
at ±¥. Also, if x1, ..., xN are the eigenvalues of H
then we will regard PNb,V as a function of the x1,..., xN. Thus,
PNb,V(x1, ..., xN)
N Õ
i=1
dxi = (ZNb,V)-1exp(-Nb
N å
i=1
V(x))
Õ
1 £ i < j £ N
|xi - xj|b
N Õ
i=1
dxi,
where ZNb,V is once again a
normalization constant. Note that the collection of Hermitian
matrices together with this new measure is not the GUE. The
function PNb,V(x1, ..., xN) is the Probability
Density of finding N eigenvalues around the points x1, ...,xN. Suppose now that we only want the probability density of
finding n eigenvalues around the points x1, ..., xn. This is
given by the n-point Correlation Function
rn,Nb,V(x1, ..., xn) =
N!
(N-n)!
ó õ
RN-n
PNb,V(x1, ...,xN)
N Õ
i=n+1
dxi.
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