  We introduce and analyze $d$ dimensional Coulomb gases with random charge
distribution and general external confining potential. We show that these gases
satisfy a large deviations principle. The analysis of the minima of the rate
function (which is the leading term of the energy) reveals that at equilibrium,
the particle distribution is a generalized circular law (i.e. with spherical
support but non-necessarily uniform distribution). In the classical
electrostatic external potential, there are infinitely many minimizers of the
rate function. The most likely macroscopic configuration is a disordered
distribution in which particles are uniformly distributed (for $d=2$, the
circular law), and charges are independent of the positions of the particles.
General charge-dependent confining potentials unfold this degenerate situation:
in contrast, the particle density is not uniform, and particles spontaneously
organize according to their charge. In that picture the classical electrostatic
potential appears as a transition at which order is lost. Sub-leading terms of
the energy are derived: we show that these are related to an operator,
generalizing the Coulomb renormalized energy, which incorporates the
heterogeneous nature of the charges. This heterogeneous renormalized energy
informs us about the microscopic arrangements of the particles, which are
non-standard, strongly depending on the charges, and include progressive and
irregular lattices.
