Determinantal point processes associated with the Bochner-Schr\"odinger operator

TL;DR

This paper analyzes the spectral properties of the Bochner-Schr"odinger operator on line bundles over manifolds, establishing asymptotic behavior of associated determinantal point processes as the tensor power increases.

Contribution

It introduces a new connection between spectral projections of the operator and determinantal point processes, with asymptotic results for large tensor powers.

Findings

Spectral projections relate to determinantal point processes.

Asymptotic behavior of linear statistics is characterized.

Law of large numbers and CLT established for empirical measures.

Abstract

We consider the Bochner-Schr\"odinger operator $H_{p}=\frac 1p\Delta^{L^p}+V$ on tensor powers $L^p$ of a Hermitian line bundle $L$ on a Riemannian manifold $X$ of bounded geometry under the assumption of non-degeneracy of the curvature form of $L$. For large $p$, the spectrum of $H_p$ asymptotically coincides with the union $\Sigma$ of all local Landau levels of the operator at the points of $X$. We study the determinantal point process on $X$ associated with the spectral projection of $H_p$ corresponding to an interval $I=(\alpha,\beta)$ such that $\alpha,\beta\not \in \Sigma$ and compute the asymptotics of its linear statistics as $p$ goes to infinity. When $X$ is compact, this implies the law of large numbers and central limit theorem for the corresponding empirical measures.