Discrepancy of determinantal point processes on compact, connected two-point homogeneous spaces

TL;DR

This paper investigates the discrepancy of determinantal point processes on compact, connected two-point homogeneous spaces, providing new bounds that extend previous results from spheres to a broader class of manifolds.

Contribution

It derives general upper bounds for the discrepancy of homogeneous determinantal point processes on all such spaces, including sharper bounds for specific ensembles.

Findings

Discrepancy of N points is O((N^{1-1/D})^{1/2} log N) with high probability for harmonic ensemble.

Sharper discrepancy bounds are obtained for the projective ensemble on complex projective spaces.

Results extend known discrepancy estimates from spheres to all compact, connected two-point homogeneous spaces.

Abstract

We study the $L^{\infty}$ discrepancy of point sets generated by determinantal point processes on all compact, connected two-point homogeneous spaces, namely spheres and projective spaces. Using concentration inequalities and variance estimates for the number of points in metric balls, we derive general upper bounds for the discrepancy of homogeneous determinantal point processes. In the particular case of the harmonic ensemble, we show that the discrepancy of $N$ points is $O((N^{1-1/D})^{1/2}\log N)$ with high probability, where $D$ denotes the real dimension of the manifold. For the projective ensemble on $\mathbb{CP}^d$, we obtain the sharper bound $O((N^{1-1/D}\log N)^{1/2})$. These results extend previously known discrepancy estimates for determinantal point processes on the sphere to all compact, connected two-point homogeneous spaces.