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Riesz Energy, $L^2$ Discrepancy, and Optimal Transport of Determinantal Point Processes on the Sphere and the Flat Torus

Bence Borda, Peter Grabner, Ryan W. Matzke

Abstract

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 $(\log N)^{1/2}$.

Riesz Energy, $L^2$ Discrepancy, and Optimal Transport of Determinantal Point Processes on the Sphere and the Flat Torus

Abstract

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 and the flat torus , and the so-called spherical ensemble on , which originates in random matrix theory. We extend results of Beltrán, Marzo and Ortega-Cerdà on the Riesz -energy of the harmonic ensemble to the nonsingular regime , and as a corollary find the expected value of the spherical cap discrepancy via the Stolarsky invariance principle. We find the expected value of the discrepancy with respect to axis-parallel boxes and Euclidean balls of the harmonic ensemble on . We also show that the spherical ensemble and the harmonic ensemble on and with points attain the optimal rate in expectation in the Wasserstein metric , in contrast to i.i.d. random points, which are known to lose a factor of .
Paper Structure (17 sections, 11 theorems, 143 equations)

This paper contains 17 sections, 11 theorems, 143 equations.

Table of Contents

  1. Introduction
  2. Results
  3. Determinantal point processes
  4. Riesz energy on the sphere
  5. $L^2$ discrepancy on the sphere
  6. $L^2$ discrepancy on the complex projective space
  7. $L^2$ discrepancies on the torus
  8. Wasserstein metric
  9. Riesz energy on $\mathbb{S}^d$
  10. $L^2$ discrepancies on $\mathbb{T}^d$
  11. Periodic $L^2$ discrepancy
  12. Ball $L^2$ discrepancy
  13. Wasserstein metric
  14. Harmonic ensemble on $\mathbb{S}^2$
  15. Harmonic ensemble on $\mathbb{T}^2$
  16. ...and 2 more sections

Key Result

Theorem 1

The harmonic ensemble $X=X(L,\mathbb{S}^d)=\{ X_1, X_2, \ldots, X_N \}$ satisfies with implied constants depending only on $s$ and $d$, where and

Theorems & Definitions (25)

  • Definition 1
  • Remark 1
  • Definition 2
  • Theorem 1
  • Corollary 2
  • Corollary 3
  • Theorem 4
  • Theorem 5
  • Theorem 6
  • Theorem 7
  • ...and 15 more