  This work concerns 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 behavior of outliers for two classes of point processes: Coulomb
gases at determinantal inverse temperature confined by a regular background,
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.
