Two short pieces around the Wigner problem

We revisit the classic Wigner semi-circle from two different angles. One consists in studying the Stieltjes transform directly on the real axis, which does not converge to a fixed value but follows a Cauchy distribution that depends on the local eigenvalue density. This result was recently proven by Aizenman and Warzel for a wide class of eigenvalue distributions. We shed new light onto their result using a Coulomb gas method. The second angle is to derive a Langevin equation for the full (matrix) resolvent, extending Dyson’s Brownian motion framework. The full matrix structure of this equation allows one to recover known results on the overlaps between the eigenvectors of a fixed matrix and its noisy counterpart, in particular in the ‘spiked’ case, for which we formulate a new conjecture.