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 mean matrix , which has matrix
rank one and nonzero eigenvalue. 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.
