  We give a potential-theoretic characterization of priors which have the
property that the corresponding Coulomb gas is "well-behaved" and similarly for
more general Riesz gases. This means that the laws of the empirical measures of
the corresponding random point process satisfy a Large Deviation Principle with
a rate functional which depends continuously on the temperature, in the sense
of Gamma-convergence. Equivalently, there is no zeroth-order phase transition
at zero temperature, in the mean field regime. This is shown to be the case for
the Hausdorff measure on a compact Lipschitz hypersurface, as well as Lesbesgue
measure on a bounded Lipschitz domain. We also provide constructions of priors,
absolutely continuous with respect to Lebesgue measure on a smoothly bounded
domain, such that the corresponding 2d Coulomb exhibits a zeroth-order phase
transition. This is based on relations to Ullman's criterion in the theory of
orthogonal polynomials and Bernstein-Markov inequalities.
