Let 
be a real symmetric
matrix of large order 
having random elements 
that for 
are independently distributed with equal densities,
equal second moments 
, and 
th moments bounded by constants 
independent of 
, 
, and 
. Further, let 
be the number of eigenvalues
of 
that lie in the interval 
for real 
. Then
![lim_(N->infty)(E[S])/N=1/(2pim^2)int_alpha^betasqrt(4m^2-x^2)dx,](/images/equations/WignersSemicircleLaw/NumberedEquation1.svg) |
where ![E[Y]](/images/equations/WignersSemicircleLaw/Inline15.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.
The distribution of eigenvalues of a symmetric random matrix with entries chosen from a standard
normal distribution is illustrated above for a random 
matrix.
Note that a large real symmetric matrix with random entries taken from a uniform distribution also obeys the semicircle law with the exception that it also possesses exactly one large eigenvalue.
