Eigenvalue Distributions of Large Hermitian Matrices; Wigner’s Semi-circle Law and a Theorem of Kac, Murdock, and Szegij

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 Szego. A new and more elementary proof of the latter is also given. The arguments use only direct L* estimates, rather than the transform methods or moment calculations employed previously.