Macroscopic and edge behavior of a planar jellium

Abstract

We consider a planar Coulomb gas in which the external potential is generated by a smeared uniform background of opposite-sign charge on a disk. 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 a 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 disk 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 upon 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.