• DOI:10.1214/21-ejp700
• Corpus ID: 244876623

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

@article{Berman2018PriorsLT,
  title={Priors leading to well-behaved Coulomb and Riesz gases versus zeroth-order phase transitions – a potential-theoretic characterization},
  author={Robert J Berman},
  journal={Electronic Journal of Probability},
  year={2018}
}

• R. Berman
• Published 26 November 2018
• Mathematics
• Electronic Journal of Probability

We give a potential-theoretic characterization of measures µ 0 which have the property that the Coulomb gas, defined with respect to the prior µ 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 µ 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.