Determinantal point processes associated with Bergman kernels: construction and limit theorems

TL;DR

This paper investigates determinantal point processes derived from Bergman kernels on complex manifolds, establishing their scaling limits, convergence properties, and large deviation principles, thus connecting complex geometry with random matrix theory.

Contribution

It introduces a new class of determinantal processes linked to Bergman kernels, analyzes their asymptotic behavior, and relates them to complex Monge-Ampère equations and Mabuchi functional.

Findings

Scaling limit is the multidimensional infinite Ginibre ensemble.

Empirical measures converge to an equilibrium measure.

Large deviation principle with Mabuchi functional as rate function.

Abstract

We study determinantal point processes whose correlation kernel is the Bergman kernel of a high power of a positive Hermitian holomorphic line bundle over a compact complex manifold. We construct such processes in analogy to the orthogonal ensembles in random matrix theory, where the correlation kernel is the famous Christoffel-Darboux kernel. Using a near-diagonal expansion of the Bergman kernel, we prove that the scaling limit of these point processes is given by a multidimensional generalization of the infinite Ginibre ensemble. As an application, we obtain a convergence in probability of their empirical measures to an equilibrium measure related to the complex Monge-Amp\`ere equations. We finally establish a large deviation principle for weighted versions of these processes, whose rate function is the Legendre-Fenchel transform of the Mabuchi functional.