Large deviations for Gibbs measures with singular Hamiltonians and emergence of Kahler-Einstein metrics

Abstract

In the present paper and the companion paper [9] a probabilistic (statistical-mechanical) approach to the construction of canonical metrics on a 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 Kahler-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 [9] 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 Kahler-Einstein metrics with positive Ricci curvature is developed.

The present paper, together with arXiv:1307.3634v2, supersedes arXiv:1307.3634v1

References in corpus (4)

• Numerical Kaehler-Einstein metric on the third del Pezzo
• Laplacian comparison for Alexandrov spaces
• Statistical mechanics of permanents, real-Monge-Ampere equations and optimal transport
• Geometry and large N limits in Laughlin states

Cited by in corpus (8)

• Quantitative estimate of propagation of chaos for stochastic systems with $W^{-1,\infty }$ kernels
• Coulomb and Riesz gases: The known and the unknown
• Large deviations for empirical measures of mean field Gibbs measures
• Mutual Asymptotic Fekete Sequences
• Emergent complex geometry
• On large deviation principles and the Monge--Ampère equation (following Berman, Hultgren)
• On large deviations for Gibbs measures, mean energy and Gamma-convergence
• Expected centre of mass of the random Kodaira embedding