Riesz Energy, $L^2$ Discrepancy, and Optimal Transport of Determinantal Point Processes on the Sphere and the Flat Torus
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2023
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| author | Borda, Bence Grabner, Peter Matzke, Ryan W. |
| author_facet | Borda, Bence Grabner, Peter Matzke, Ryan W. |
| contents | 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 $\mathbb{S}^d$ and the flat torus $\mathbb{T}^d$, and the so-called spherical ensemble on $\mathbb{S}^2$, 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 $\mathbb{T}^d$. We also show that the spherical ensemble and the harmonic ensemble on $\mathbb{S}^2$ and $\mathbb{T}^2$ 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 $(\log N)^{1/2}$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2308_06216 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Riesz Energy, $L^2$ Discrepancy, and Optimal Transport of Determinantal Point Processes on the Sphere and the Flat Torus Borda, Bence Grabner, Peter Matzke, Ryan W. Classical Analysis and ODEs Mathematical Physics Probability 60G55, 31C12, 49Q22, 11K38 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 $\mathbb{S}^d$ and the flat torus $\mathbb{T}^d$, and the so-called spherical ensemble on $\mathbb{S}^2$, 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 $\mathbb{T}^d$. We also show that the spherical ensemble and the harmonic ensemble on $\mathbb{S}^2$ and $\mathbb{T}^2$ 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 $(\log N)^{1/2}$. |
| title | Riesz Energy, $L^2$ Discrepancy, and Optimal Transport of Determinantal Point Processes on the Sphere and the Flat Torus |
| topic | Classical Analysis and ODEs Mathematical Physics Probability 60G55, 31C12, 49Q22, 11K38 |
| url | https://arxiv.org/abs/2308.06216 |