Researcher ORCID Identifier
Open Access Senior Thesis
Bachelor of Arts
A combinatorial proof of Wigner’s semicircle law for the Gaussian Unitary Ensemble (GUE) is presented using techniques from free probability. Motivating examples taken from the symmetric Bernoulli ensemble and the GUE show the distribution of eigenvalues of sample n x n matrices approaching Wigner’s semicircle as n get large. The concept of crossing and non-crossing pairings is developed, along with proofs of Wick’s Formula for real and complex Gaussians. It is shown that Wigner’s semicircle distribution has moments given by the Catalan numbers. Wick’s Formula and several additional lemmas (proved in sequence) lead to a "method of moments" proof that the expectation of powers of eigenvalues (spectra) of large random matrices from the GUE converge in expectation to the Catalan numbers, proving Wigner’s semicircle law in expectation.
Wolf, Vanessa, "Random Matrix Theory: A Combinatorial Proof of Wigner's Semicircle Law" (2021). Scripps Senior Theses. 1683.