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 ${\exp }_p:T_p^{\ast }M\to M$ under a suitable scaling converges weakly to the universal determinantal point process on $T_p^{\ast }M$ as $\lambda \to \infty$.

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