Determinantal Point Processes Associated with the Bochner–Schrödinger Operator

We consider the Bochner–Schrödinger operator Hₚ= 1pΔLᵖ+V on tensor powers Lᵖ of a Hermitian line bundle L on a Riemannian manifold X of bounded geometry under the assumption of nondegeneracy of the curvature form of L . For large p , the spectrum of Hₚ asymptotically coincides with the union Σ 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 Hₚ corresponding to an interval I=(α,β) such that α,β ∈ Σ 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.