  We consider a planar Coulomb gas in which the external potential is generated
by a smeared uniform background of opposite-sign charge on a disc. This model
can be seen as a two-dimensional Wigner jellium, not necessarily charge
neutral, and with particles allowed to exist beyond the support of the smeared
charge. The full space integrability condition requires low enough temperature
or high enough total smeared charge. This condition does not allow at the same
time, total charge neutrality and determinantal structure. The model shares
similarities with both the complex Ginibre ensemble and the
Forrester--Krishnapur spherical ensemble of random matrix theory. In
particular, for a certain regime of temperature and total charge, the
equilibrium measure is uniform on a disc as in the Ginibre ensemble, while the
modulus of the farthest particle has heavy-tailed fluctuations as in the
Forrester--Krishnapur spherical ensemble. We also touch on a higher temperature
regime producing a crossover equilibrium measure, as well as a transition to
Gumbel edge fluctuations. More results in the same spirit on edge fluctuations
are explored by the second author together with Raphael Butez.
