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

Sharp asymptotics for Toeplitz determinants, fluctuations and the gaussian free field on a Riemann surface

We consider canonical determinantal random point processes with N particles on a compact Riemann surface X defined with respect to the constant curvature metric. In the higher genus (hyperbolic) cases these point processes may be defined in terms of automorphic forms. We establish strong exponential concentration of measure type properties involving Dirichlet norms of linear statistics. This gives an optimal Central Limit Theorem (CLT), saying that the fluctuations of the corresponding empirical measures converge, in the large N limt, towards the Laplacian of the Gaussian free field on X in the strongest possible sense. The CLT is also shown to be equivalent to a new sharp strong Szeg\"o type theorem for Toeplitz determinants in this context. One of the ingredients in the proofs are new Bergman kernel asymptotics providing exponentially small error terms in a constant curvature setting.Comment: v1: 15 pages, v2: 21 pages, added new Bergman kernel asymptotics with exponentially small error term