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arXiv · 1811.10249

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

Robert J. Berman
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Abstract

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.

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Robert J. Berman. 2018-11-26. Priors leading to well-behaved Coulomb and Riesz gases versus zeroth-order phase transitions -- a potential-theoretic characterization. https://arxiv.org/abs/1811.10249

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