Abstract
We give a potential-theoretic characterization of measures μ0 which have the property that the Coulomb gas, defined with respect to the prior μ0, 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 μ0, 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.
Highlights
In broad terms, the main aim of the present work is to study the interplay between fine potential-theoretic properties of a measure μ0 in random measure on (Rd) and properties of the corresponding Coulomb gas in Rd, in the mean-field regime
Priors leading to well-behaved Coulomb, Riesz gases vs zeroth-order phase transitions corresponding Coulomb gas is “well-behaved at zero-temperature”, which is equivalent to the absence of a zeroth-order phase transition
Let μ0 be a measure on Rd which does not charge polar subsets and assume that the support S0 of μ0 is compact and locally regular
Summary
The main aim of the present work is to study the interplay between fine potential-theoretic properties of a measure μ0 in Rd (the “prior”) and properties of the corresponding Coulomb gas in Rd, in the mean-field regime. If ν is strongly determining and does not charge polar subsets the support of ν is automatically locally regular (see Section 2.7 for the potential-theoretic notions of regularity). Theorem 1.1 appears to be new even in the simplest case when S0 is an interval in R ⊂ R2, which is the classical setting where the notion of determining measures was first introduced by Ullman (as discussed in Section 1.3 below). In this special case a measure μ0 is determining iff it is strongly determining
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