  This work treats weakly confined particle systems in the plane, characterized
by a large number of outliers away from a droplet where the bulk of the
particles accumulate in the many-particle limit. We are interested in the
asymptotic behaviour of outliers for two families of point processes: Coulomb
gases confined by regular background measures at determinantal inverse
temperature, and a class of random polynomials. We observe that the limiting
outlier process only depends on the shape of the uncharged region containing
them, and the global net excess charge.
  In particular, for a determinantal Coulomb gas confined by a sufficiently
regular background measure, the outliers in a simply connected uncharged region
converge to the corresponding Bergman point process. For a finitely connected
uncharged region $\Omega$, a family of limiting outlier processes arises,
indexed by the (Pontryagin) dual of the fundamental group of $\Omega$.
Moreover, the outliers in different uncharged regions are asymptotically
independent, even if the regions have common boundary points. The latter result
is a manifestation of screening properties of the particle system.
