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 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.

Ensemble Equivalence for Mean Field Models and Plurisubharmonicity

“…It then follows from general principles that F(β) is differentiable with derivative at β given by e(β) := E(μ β ), where μ β is the unique minimizer of F β . Indeed, this follows from the general statement in appendix of [9] (see [7]). All that remains is thus to verify that lim β→0 e(β) = e 0 .…”

“…We then deduce that thermodynamic equivalence of ensembles holds for any energy level e in ]e min , e max [ using a general result (Theorem 5.4), saying that for a general lower semi-continuous convex energy functionals E(μ) and prior μ 0 thermodynamic equivalence of ensembles holds in the low-energy region ]e min , e 0 [ iff E(μ) and μ 0 satisfy a certain compatibility property (the "energy approximation property"). This property has previously appeared in connection with the study of large deviation principles for the corresponding canonical ensembles at positive inverse temperatures β [7,17].…”

“…However, as shown in [7] the latter convergence holds for all V ∈ C 0 (X ) iff μ 0 has the approximation property (briefly, the point is that the convergence in question is, since E(μ) is convex equivalent to the -convergence of E(μ) − S(μ)/β toward E(μ), which, in turn, is equivalent to the energy approximation property of μ 0 ).…”

“…But since e(β) is decreasing, this follows readily from the lower semi-continuity of E(μ) (see [7]). To prove that that S(e) is continuous on ]e min , e 0 ] it will be enough, by the previous step, to show that F * (e) is continuous on ]e min , e 0 ] and F * (e 0 ) = 0.…”

“…Remark 3.5. This property was introduced in the context of large deviation theory in [17] and studied from a potential-theoretic point of view in [7] (see the discussion in the end of Sect. 5.2).…”