University Honors

Around 1920, Wishart began researching how different sets of data could be connected. He focusedon the eigenvalues of matrices, which are special numbers that can inform us about the structureof the data. This work laid the foundation for Random Matrix Theory, a field that combines ideasfrom linear algebra and probability.In the 1950s, Wigner, an American physicist, used Random Matrix Theory to study the behaviorof eigenvalues in large matrices. He found that as the size of the matrices became very large, thedistribution of eigenvalues tended to form a semicircle shape. This unexpected result, known asWigner’s Semicircle Law, showed that the distribution of eigenvalues was not normal, but insteadresembled a semicircle.This project aims to prove Wigner’s Semicircle Law using concepts from probability, linear al-gebra, and rigorous mathematical proofs. The goal is to provide a clear explanation of the proofand to understand the reasoning in why this happens. The aim of demonstrating the proof of thesemicircle law is to present the work in a manner that is understandable to a wide variety of people,even those without a background in mathematics. By doing this, we hope to understand why thesemicircle law holds and whether it remains true under different assumptions.