Local universality of determinantal point processes on Riemannian manifolds

Abstract

We consider the Laplace-Beltrami operator $\Delta_{g}$ on a smooth, compact Riemannian manifold $(M,g)$ and the determinantal point process $\mathcal{X}_{\lambda}$ on $M$ associated with the spectral projection of $-\Delta_{g}$ onto the subspace corresponding to the eigenvalues up to $\lambda^{2}$. We show that the pull-back of $\mathcal{X}_{\lambda}$ by the exponential map $\exp_{p} : T_{p}^{*}M \to M$ under a suitable scaling converges weakly to the universal determinantal point process on $T_{p}^{*} M$ as $\lambda \to \infty$.