Determinantal point processes and fermions on complex manifolds: Bulk universality

TL;DR

This paper studies determinantal point processes on complex manifolds, showing bulk universality and Gaussian fluctuations, with applications to random matrices, Coulomb gases, and fermions.

Contribution

It establishes bulk universality and Gaussian fluctuation results for determinantal processes on complex manifolds, extending known results to higher dimensions and more general geometric settings.

Findings

Empirical measures converge to pluripotential equilibrium measure.

Bulk correlation functions are universal and relate to higher-dimensional Ginibre ensemble.

Fluctuations are asymptotically Gaussian and described by a Gaussian free field.

Abstract

We consider determinantal point processes on a compact complex manifold X in the limit of many particles. The correlation kernels of the processes are the Bergman kernels associated to a a high power of a given Hermitian holomorphic line bundle L over X. The empirical measure on X of the process, describing the particle locations, converges in probability towards the pluripotential equilibrium measure, expressed in term of the Monge-Amp\`ere operator. The asymptotics of the corresponding fluctuations in the bulk are shown to be asymptotically normal and described by a Gaussian free field and applies to test functions (linear statistics) which are merely Lipschitz continuous. Moreover, a scaling limit of the correlation functions in the bulk is shown to be universal and expressed in terms of (the higher dimensional analog of) the Ginibre ensemble. This geometric setting applies in particular to normal random matrix ensembles, the two dimensional Coulomb gas, free fermions in a strong magnetic field and multivariate orthogonal polynomials.