Wigner's Semicircle Law
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
(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.
REFERENCES:
Arnold, L. "On Wigner's Semicircle Law for the Eigenvalues of Random Matrices." Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 19, 191-198, 1971.
Bai, Z. D. and Yin, Y. Q. "Convergence to the Semicircle Law." Ann. Probab. 16, 863-875, 1988.
Götze, F. and Tikhomirov, A. "Rate of Convergence to the Semi-Circular Law." Probab. Theory Related Fields 127, 228-276, 2003.
Kiessling, M. K.-H. and Spohn, H. "A Note on the Eigenvalue Density of Random Matrices." Comm. Math. Phys. 199, 683-695, 1999.
Ryan, Ø. "On the Limit Distributions of Random Matrices with Independent or Free Entries." Comm. Math. Phys. 193, 595-626, 1998.
Voiculescu, D. "Limit Laws for Random Matrices and Free Products." Invent. Math. 104, 201-220, 1991.
Wigner, E. "Characteristic Vectors of Bordered Matrices with Infinite Dimensions." Ann. of Math. 62, 548-564, 1955.
Wigner, E. "On the Distribution of the Roots of Certain Symmetric Matrices." Ann. of Math. 67, 325-328, 1958.
