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

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.