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.