Riesz Energy, L^2 Discrepancy, and Optimal Transport of Determinantal Point Processes on the Sphere and the Flat Torus
arXiv (Cornell University)(2023)
摘要
Determinantal point processes exhibit an inherent repulsive behavior, thus
providing examples of very evenly distributed point sets on manifolds. In this
paper, we study the so-called harmonic ensemble, defined in terms of Laplace
eigenfunctions on the sphere 𝕊^d and the flat torus 𝕋^d,
and the so-called spherical ensemble on 𝕊^2, which originates in
random matrix theory. We extend results of Beltrán, Marzo and Ortega-Cerdà
on the Riesz s-energy of the harmonic ensemble to the nonsingular regime
s<0, and as a corollary find the expected value of the spherical cap L^2
discrepancy via the Stolarsky invariance principle. We find the expected value
of the L^2 discrepancy with respect to axis-parallel boxes and Euclidean
balls of the harmonic ensemble on 𝕋^d. We also show that the
spherical ensemble and the harmonic ensemble on 𝕊^2 and
𝕋^2 with N points attain the optimal rate N^-1/2 in
expectation in the Wasserstein metric W_2, in contrast to i.i.d. random
points, which are known to lose a factor of (log N)^1/2.
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关键词
riesz energy,optimal transport,point processes,sphere
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