Priors leading to well-behaved Coulomb and Riesz gases versus zeroth-order phase transitions – a potential-theoretic characterization

Abstract

We give a potential-theoretic characterization of measures ${\mathrm{\mu }}_0$ which have the property that the Coulomb gas, defined with respect to the prior ${\mathrm{\mu }}_0$, 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 ${\mathrm{\mu }}_0$, 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.