The distribution of eigenvalues of random matrices is a central idea in random matrix theory. Universality is a phenomenon which holds for a wide class of random variables, usually motivated by normal random variables. We will look at Wigner's famous semicircle law, the circular law, the elliptic law, and results on the distributions of eigenvalues on the unit circle for probability measures on compact matrix groups. No prior knowledge on probability or random matrix theory will be assumed. Pictures will be included.
Universality, Eigenvalues of Random Matrices, and Circles