Determinantal structure of the GUE

Andrew Lin
Fri, Mar 1 2024, 4:00pm

Abstract: This quarter, we've explored the empirical distribution of the eigenvalues for general Wigner matrices, showing convergence in distribution to the semicircle law for Hermitian matrices. As Christian described, the special case with Gaussian entries allows us to say more and get an explicit formula for the eigenvalue density. It turns out this formula (involving the Vandermonde determinant) can be rewritten using Hermite polynomials to get joint densities of k eigenvalues at a time in terms of determinants, leading to limiting formulas involving the sine kernel sin(x-y)/(x-y) and asymptotics of eigenvalues both in the "bulk" of the semicircle and at the edge of the spectrum. In this talk, I'll set up some of the fundamental objects towards proving these results, following some parts of Chapter 3 in Anderson-Guionnet-Zeitouni.