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

We give a potential-theoretic characterization of measures μ0 which have the property that Coulomb gas, defined with respect to prior μ0, is “well-behaved” and similarly for more general Riesz gases. This means laws empirical corresponding random point process satisfy Large Deviation Principle rate functional depends continuously on temperature, in sense Gamma-convergence. Equivalently, there no zeroth-order phase transition at zero mean field regime. shown be case Hausdorff measure compact Lipschitz hypersurface, as well Lesbesgue bounded domain. also provide constructions priors absolutely continuous Lebesgue smoothly domain, such 2d exhibits transition. based relations Ullman’s criterion theory orthogonal polynomials Bernstein-Markov inequalities.