Eigenvalue distributions of large Hermitian matrices; Wigner's semi-circle law and a theorem of Kac, Murdock, and Szegö

Abstract

Wigner's semi-circle law describes the eigenvalue distribution of certain large random Hermitian matrices. A new proof is given for the case of Gaussian matrices, that involves reducing a random matrix to tridiagonal form by a method that is well known as a technique for numerical computation of eigenvalues. The result is a generalized Toeplitz matrix whose eigenvalue distribution can be found using a theorem of Kac, Murdock, and Szegö. A new and more elementary proof of the latter is also given. The arguments use only direct L 2 estimates, rather than the transform methods or moment calculations employed previously.