Permanental Point Processes on Real Tori, Theta Functions and Monge–Ampère Equations

Hultgren, Jakob

doi:10.5802/afst.1592

Cited by 4 publications

(6 citation statements)

References 25 publications

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“…It turns out that the validity of the conjecture above would follow from the existence of the corresponding mean energyĒ(µ), for any volume form µ (see Problem 4.10). This is shown precisely as in the setting of the real Monge-Ampère operator considered in [8,42] where the analog of the previous conjecture was established using permanents as a replacements of the determinants appearing in the present setting. In particular, when X is a Calabi-Yau manifold, i.e.…”

Section: 4supporting

confidence: 73%

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Large Deviations for Gibbs Measures with Singular Hamiltonians and Emergence of Kähler–Einstein Metrics

Berman

2017

Commun. Math. Phys.

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Abstract:In the present paper and the companion paper (Berman, Kähler-Einstein metrics, canonical random point processes and birational geometry. arXiv:1307.3634, 2015) a probabilistic (statistical-mechanical) approach to the construction of canonical metrics on complex algebraic varieties X is introduced by sampling "temperature deformed" determinantal point processes. The main new ingredient is a large deviation principle for Gibbs measures with singular Hamiltonians, which is proved in the present paper. As an application we show that the unique Kähler-Einstein metric with negative Ricci curvature on a canonically polarized algebraic manifold X emerges in the many particle limit of the canonical point processes on X. In the companion paper (Berman in 2015) the extension to algebraic varieties X with positive Kodaira dimension is given and a conjectural picture relating negative temperature states to the existence problem for Kähler-Einstein metrics with positive Ricci curvature is developed.

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Section: 4supporting

confidence: 73%

“…In particular, relations to algebro-geometric stability properties, as in the Yau-Tian-Donaldson conjecture are described in [9]. See also [8,42] for connections to optimal transport in the real setting (corresponding to the case when X is toric and abelian variety, respectively) and [6] for connections to physics.…”

Section: Introductionmentioning

confidence: 99%

Large Deviations for Gibbs Measures with Singular Hamiltonians and Emergence of Kähler–Einstein Metrics

Berman

2017

Commun. Math. Phys.

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“…However, as mentioned above the geometric properties of solutions to equation (9) have very recently been studied by Klartag and Kolesnikov in [12]. Moreover, when M = R n , (9) has been studied as a twisted Kähler-Einstein equation on a corresponding toric manifold (see [21,2]) and when M is the real torus with the standard affine structure (9) has been studied as an analog of a twisted singular Kähler-Einstein equation in [11]. In the case when λ > 0 we will also show uniqueness of solutions to (9).…”

Section: Further Resultsmentioning

confidence: 99%

“…First of all, note that the fibers of the quotient map are affine submanifolds of K * isomorphic to R. Moreover, there is a global affine trivialization of K * over Ω * . To see this, recall that by Remark 7 K * is isomorphic to a subset of (R n ) * × R. The action on K * given by (11) extends to all of (R n ) * × R where it is given by (a, b) → (a, b + C). In particular, the quotient map is the same as the projection map on the first factor.…”

Section: Remarkmentioning

confidence: 99%

An Optimal Transport Approach to Monge–Ampère Equations on Compact Hessian Manifolds

Hultgren

Önnheim

2018

J Geom Anal

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In this paper we consider Monge-Ampère equations on compact Hessian manifolds, or equivalently Monge-Ampère equations on certain unbounded convex domains Ω ⊆ R n , with a periodicity constraint given by the action of an affine group. In the case where the affine group action is volume-preserving, i.e., when the manifold is special, the solvability of the corresponding Monge-Ampère equation was established using the continuity method in [6]. In the general case we set up a variational framework involving certain dual manifolds and a generalization of the classical Legendre transform. We give existence and uniqueness results, elaborate on connections to optimal transport and quasi-periodic tilings of convex domains.

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“…It would be interesting establish quantitative rates in the case when β < ∞. In another direction, in the case when (X, L) is an abelian variety (i.e X is a compact complex torus) a real analog of Theorem 5.7, which holds also for negative β, has been established in [74].…”

Section: Large Deviations For Vandermonde Determinantsmentioning

confidence: 99%

Statistical mechanics of interpolation nodes, pluripotential theory and complex geometry

Berman¹

2019

Ann. Polon. Math.

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This is mainly a survey, explaining how the probabilistic (statistical mechanical) construction of Kähler-Einstein metrics on compact complex manifolds, introduced in a series of works by the author, naturally arises from classical approximation and interpolation problems in C n . A fair amount of background material is included. Along the way, the results are generalized to the non-compact setting of C n . This yields a probabilistic construction of Kähler solutions to Einstein's equations in C n , with cosmological constant −β, from a gas of interpolation nodes in equilibrium at positive inverse temperature β. In the infinite temperature limit, β → 0, solutions to the Calabi-Yau equation are obtained. In the opposite zero-temperature the results may be interpreted as "transcendental" analogs of classical asymptotics for orthogonal polynomials, with the inverse temperature β playing the role of the degree of a polynomial.

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Permanental Point Processes on Real Tori, Theta Functions and Monge–Ampère Equations

Abstract: Inspired by constructions in complex geometry we introduce a thermodynamic framework for Monge-Ampère equations on real tori. We show convergence in law of the associated point processes and explain connections to complex Monge-Ampère equations and optimal transport.

Cited by 4 publications

References 25 publications

Large Deviations for Gibbs Measures with Singular Hamiltonians and Emergence of Kähler–Einstein Metrics

Large Deviations for Gibbs Measures with Singular Hamiltonians and Emergence of Kähler–Einstein Metrics

An Optimal Transport Approach to Monge–Ampère Equations on Compact Hessian Manifolds

Statistical mechanics of interpolation nodes, pluripotential theory and complex geometry

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