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We introduce and analyze $d$ dimensional Coulomb gases with random chargedistribution and general external confining potential. We show that these gasessatisfy a large deviations principle. The analysis of the minima of the ratefunction (which is the leading term of the energy) reveals that at equilibrium,the particle distribution is a generalized circular law (i.e. with sphericalsupport but non-necessarily uniform distribution). In the classicalelectrostatic external potential, there are infinitely many minimizers of therate function. The most likely macroscopic configuration is a disordereddistribution in which particles are uniformly distributed (for $d=2$, thecircular 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 spontaneouslyorganize according to their charge. In that picture the classical electrostaticpotential appears as a transition at which order is lost. Sub-leading terms ofthe energy are derived: we show that these are related to an operator,generalizing the Coulomb renormalized energy, which incorporates theheterogeneous nature of the charges. This heterogeneous renormalized energyinforms us about the microscopic arrangements of the particles, which arenon-standard, strongly depending on the charges, and include progressive andirregular lattices.

The heterogeneous gas with singular interaction: Generalized circular law and heterogeneous renormalized energy