Wigner's semicircle law describes the limiting eigenvalue distribution of a large real symmetric matrix
with independent centered entries. Let be such a matrix of order , with the entries for having mean 0, common variance with , and th absolute moments bounded
by constants independent of , , and . Further, let be the number of eigenvalues
of
that lie in the interval for . Then
where
denotes the expectation value of (Wigner 1955, 1958). This law was first observed by Wigner
(1955) for certain special classes of random matrices
arising in quantum mechanical investigations.
A centered uniform distribution, such as the uniform distribution on , also gives the semicircle law. For a noncentered distribution
with mean, writing , where is the unit matrix, isolates
the meanmatrix, which has matrix rank one
and nonzeroeigenvalue. For example, entries uniformly distributed
on
give the same centered semicircle bulk together with one eigenvalue
of order
outside that bulk.
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be such a matrix of order 
, with the entries 
for 
having mean 0, common variance 
with 
, and 
th absolute moments bounded
by constants 
independent of 
, 
, and 
. Further, let 
be the number of eigenvalues
of 
that lie in the interval 
for 
. Then
![lim_(N->infty)(E[S])/N=1/(2pim^2)int_alpha^betasqrt(4m^2-x^2)dx,](/images/equations/WignersSemicircleLaw/NumberedEquation1.svg)
![E[Y]](/images/equations/WignersSemicircleLaw/Inline16.svg)
denotes the expectation value of 
(Wigner 1955, 1958). This law was first observed by Wigner
(1955) for certain special classes of random matrices
arising in quantum mechanical investigations.
matrix.
![[-1,1]](/images/equations/WignersSemicircleLaw/Inline19.svg)
, also gives the semicircle law. For a noncentered distribution
with mean 
, writing 
, where 
is the unit matrix, isolates
the mean matrix 
, which has matrix rank one
and nonzero eigenvalue 
. For example, entries uniformly distributed
on ![[0,1]](/images/equations/WignersSemicircleLaw/Inline25.svg)
give the same centered semicircle bulk together with one eigenvalue
of order 
outside that bulk.
