Wigner Matrices and the Semicircle Law

Abstract: We prove the convergence in distribution of the empirical law of the eigenvalues of a Wigner matrix to the semicircle distribution. This is a classical result dating to the founding of the subject. The proof is by the moment method - specifically the convergence of the random moments of the empirical law to those given by the semicircle distribution. This convergence is accomplished by a combinatorial argument.