Eugene Wigner’s revolutionary vision predicted that the energy levels of large complex quantum systems exhibit a universal behavior depending only on the symmetry type of the model.
One class of such models are invariant matrix ensembles for which the eigenvalue distribution distributions are given by Coulomb-gases with potential $V$ and inverse temperature beta.
In this lecture we will sketch the proof of the universality of Coulomb gases both in the bulk and at the edges of the spectrum.
We will link the universality problem to the decay of correlation functions of Coulomb gases and show that such a decay follows from a Holder regularity of a discrete parabolic equation with random coefficients.
The parabolic regularity will be established partly using the recent argument of Caffarelli-Chan-Vasseur, which is a De Giorgi-Nash-Moser type method.
The singularities in random coefficients pose major challenges; We will use optimal level repulsion estimates in random matrices to control them.