From Monge-Ampere equations to envelopes and geodesic rays in the zero temperature limit

Let X be a compact complex manifold equipped with a smooth (but not necessarily positive) closed form theta of one-one type. By a well-known envelope construction this data determines a canonical theta-psh function u which is not two times differentiable, in general. We introduce a family of regularizations of u, parametrized by a positive number beta, defined as the smooth solutions of complex Monge-Ampere equations of Aubin-Yau type. It is shown that, as beta tends to infinity, the regularizations converge to the envelope u in the strongest possible Holder sense. A generalization of this result to the case of a nef and big cohomology class is also obtained. As a consequence new PDE proofs are obtained for the regularity results for envelopes in [14] (which, however, are weaker than the results in [14] in the case of a non-nef big class). Applications to the regularization problem for quasi-psh functions and geodesic rays in the closure of the space of Kahler metrics are given. As briefly explained there is a statistical mechanical motivation for this regularization procedure, where beta appears as the inverse temperature. This point of view also leads to an interpretation of the regularizations as transcendental Bergman metrics.Comment: 28 pages. Version 2: 29 pages. Improved exposition, references updated. Version 3: 31 pages. A direct proof of the bound on the Monge-Amp\`ere mass of the envelope for a general big class has been included and Theorem 2.2 has been generalized to measures satisfying a Bernstein-Markov propert

The Coulomb gas, potential theory and phase transitions

We give a potential-theoretic characterization of measures 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. This is shown to be the case for the Hausdorff measure on a Lipschitz hypersurface. We also provide explicit examples of measures which are absolutely continuous with respect to Lesbesgue measure, 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.Comment: v1: 40 pages. v2: 44 pages (improved exposition and sections 3.3, 3.4 added