  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 \& 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.
