A combinatorial proof of Wigner’s Semicircle Law for the Gaussian Unitary Ensemble (GUE) is presented in this report. The distribution of eigenvalues of different samples of general Wigner matrices is shown to converge to the semicircle distribution, with the aid of histograms created in Python. The type of convergence that is shown is that of the averaged moments of the eigenvalue
distribution of sample GUE matrices to the moments of the semicircle distribution, as the size of the matrices grow large. This is done by using a method known as the ‘method of moments’. The concepts of random matrices, Catalan numbers, mixed moments of standard Gaussian random variables, (non-
crossing) pairings, Wick’s formula and permutation cycles are introduced in this method. The aim of this report is to provide a detailed proof of Wigner’s Semicircle Law in expectation, understandable for bachelor level mathematics students.