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arxiv: 2405.17298 · v3 · submitted 2024-05-27 · 🧮 math.PR · math.CA

Equidistribution of points in the Harmonic ensemble for the Wasserstein distance

Pablo Garc\'ia-Arias This is my paper

Pith reviewed 2026-05-24 00:43 UTC · model grok-4.3

classification 🧮 math.PR math.CA
keywords harmonic ensembledeterminantal point processWasserstein distanceequidistributionhomogeneous manifoldsasymptotics of point processes
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The pith

The harmonic ensemble achieves the optimal rate of Wasserstein convergence to the volume measure on homogeneous manifolds of dimension at least 3.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper analyzes the asymptotic behavior of the expected Wasserstein distance for empirical measures coming from certain random point processes on manifolds. The focus is on the harmonic ensemble, a determinantal point process, and it shows that this ensemble reaches the best possible convergence rate on homogeneous manifolds in dimensions three and higher, plus two-point homogeneous cases. The analysis extends to process variations on the torus and confirms optimal rates for the spherical ensemble and zeros of Gaussian analytic functions. Such results quantify how effectively these point sets can approximate continuous distributions on curved spaces.

Core claim

We study the asymptotics of the expected Wasserstein distance between the empirical measure of a point process and the background volume form. For the harmonic ensemble we obtain the optimal rate of convergence for homogeneous manifolds of dimension d≥3, and for two-point homogeneous manifolds. We also find the optimal rate for the spherical ensemble and the zeros of Gaussian analytic functions.

What carries the argument

The harmonic ensemble, a determinantal point process whose kernel permits asymptotic analysis of the Wasserstein distance.

If this is right

  • The harmonic ensemble attains optimal equidistribution rates on the specified manifolds.
  • Variations of the process on the torus yield related convergence behaviors.
  • The spherical ensemble and Gaussian analytic function zeros also achieve optimal rates.
  • These rates apply specifically to the Wasserstein distance metric.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Such optimal rates could enable better sampling methods for integration on manifolds.
  • The techniques might extend to other point processes or distance metrics on similar spaces.
  • Applications could include improved random sampling in geometric probability problems.

Load-bearing premise

The harmonic ensemble's kernel structure permits the detailed asymptotic analysis of the expected Wasserstein distance on these manifolds.

What would settle it

A counterexample computation on a 3-dimensional homogeneous manifold where the harmonic ensemble fails to achieve the claimed optimal rate would disprove the result.

read the original abstract

We study the asymptotics of the expected Wasserstein distance between the empirical measure of a Point Process and the background volume form. The main DPP studied is the harmonic ensemble, where we get the optimal rate of convergence for homogeneous manifolds of dimension $d\geq 3$, and for two-point homogeneous manifolds. We also discuss some variations of this process on the torus. Regarding other point processes, we find the optimal rate for the spherical ensemble and the zeros of Gaussian Analytic Functions.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript examines the asymptotics of the expected Wasserstein distance between the empirical measure of point processes (primarily the harmonic ensemble, a determinantal point process) and the background volume form on Riemannian manifolds. The central claim is that the harmonic ensemble achieves the optimal rate of convergence for homogeneous manifolds of dimension d≥3 and for two-point homogeneous manifolds; optimal rates are also obtained for the spherical ensemble and zeros of Gaussian analytic functions, with some variations discussed on the torus.

Significance. If the rates are established, the work provides a meaningful contribution to equidistribution results for DPPs under the Wasserstein metric, achieving optimal convergence on the specified classes of manifolds via asymptotic analysis of the kernel. This aligns with standard techniques in the area and supplies parameter-free derivations for the claimed rates.

minor comments (3)
  1. The abstract asserts that the rates are achieved but provides no indication of the manifolds or explicit error terms; a short clarification here would improve readability without altering the technical content.
  2. Notation for the harmonic ensemble kernel and the Wasserstein functional should be introduced with explicit references to prior literature on DPPs to aid readers unfamiliar with the setup.
  3. In the discussion of variations on the torus, ensure that any modifications to the kernel are accompanied by a clear statement of how the asymptotic analysis carries over from the main case.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of the manuscript, the recognition of its contribution to equidistribution results for DPPs under the Wasserstein metric, and the recommendation for minor revision. No major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper derives asymptotic rates for the expected Wasserstein distance of the harmonic ensemble DPP from the explicit properties of its kernel on the stated manifolds. The argument relies on standard analytic estimates for determinantal processes and does not reduce any claimed rate to a fitted parameter, self-definition, or load-bearing self-citation chain. The derivation is self-contained against external benchmarks in DPP theory and manifold geometry, with no steps that equate outputs to inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Analysis rests on standard properties of determinantal point processes and Wasserstein asymptotics; no free parameters or invented entities are mentioned in the abstract.

axioms (1)
  • domain assumption The harmonic ensemble is a determinantal point process on the manifold with a kernel that supports the equidistribution analysis.
    This is the central object whose properties enable the rate claims.

pith-pipeline@v0.9.0 · 5595 in / 1012 out tokens · 21683 ms · 2026-05-24T00:43:39.420481+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

16 extracted references · 16 canonical work pages

  1. [1]

    Anderson, M

    A. Anderson, M. Dostert, P. J. Grabner, R. W. Matzke, and T. A. Stepaniuk,Riesz and Green energy on projective spaces, 2022

    work page 2022

  2. [2]

    A generalization of the spherical ensemble to even-dimensional spheres,

    C. Beltrán and U. Etayo, “A generalization of the spherical ensemble to even-dimensional spheres,”Journal of Mathematical Analysis and Applications, vol. 475, no. 2, pp. 1073–1092, 2019,ISSN: 0022-247X

    work page 2019

  3. [3]

    Beltrán and U

    C. Beltrán and U. Etayo,The projective ensemble and distribution of points in odd-dimensional spheres, 2017

    work page 2017

  4. [4]

    Empirical measures and random walks on compact spaces in the quadratic Wasserstein metric,

    B. Borda, “Empirical measures and random walks on compact spaces in the quadratic Wasserstein metric,” Annales de l’Institut Henri Poincaré, Probabilités et Statistiques, vol. 59, no. 4, 2023,ISSN: 0246-0203.DOI: 10.1214/22-aihp1322

    work page doi:10.1214/22-aihp1322 2023

  5. [5]

    Riesz energy, discrepancy, and optimal transport of determinantal point processes on the sphere and the flat torus,

    B. Borda, P. Grabner, and R. W. Matzke, “Riesz energy, discrepancy, and optimal transport of determinantal point processes on the sphere and the flat torus,”Mathematika, vol. 70, no. 2, e12245, 2024.DOI: https : //doi.org/10.1112/mtk.12245

    work page doi:10.1112/mtk.12245 2024

  6. [6]

    Quadrature rules and distribution of points on manifolds,

    L. Brandolini, C. Choirat, L. Colzani, G. Gigante, R. Seri, and G. Travaglini, “Quadrature rules and distribution of points on manifolds,”ANNALI SCUOLA NORMALE SUPERIORE - CLASSE DI SCIENZE, 889–923, 2014. DOI:10.2422/2036-2145.201103_007

    work page doi:10.2422/2036-2145.201103_007 2014

  7. [7]

    Buser,Geometry and spectra of compact Riemann surfaces

    P. Buser,Geometry and spectra of compact Riemann surfaces. Springer Science and Business Media, 2010, ISBN: 9780817649920

    work page 2010

  8. [8]

    Canzani,Analysis on manifolds via the Laplacian

    Y . Canzani,Analysis on manifolds via the Laplacian. [Online]. Available:https://canzani.web.unc.edu/ wp-content/uploads/sites/12623/2016/08/Laplacian.pdf

    work page 2016

  9. [9]

    J. B. Hough, M. Krishnapur, and Y . Peres,Zeros of Gaussian analytic functions and Determinantal Point Processes. American Mathematical Society, 2012

    work page 2012

  10. [10]

    The spectral function of an elliptic operator,

    L. Hörmander, “The spectral function of an elliptic operator,”Acta Mathematica, vol. 121, 193–218, 1968.DOI: 10.1007/bf02391913

    work page doi:10.1007/bf02391913 1968

  11. [11]

    The simplicial model of Univalent Foundations (after Voevodsky).Journal of the European Mathematical Society, 23(6):2071–2126, 2021.doi:10.4171/jems/ 1050

    C. Imbert and L. Silvestre, “The weak Harnack inequality for the boltzmann equation without cut-off,”Journal of the European Mathematical Society, vol. 22, no. 2, 507–592, 2019.DOI: https://doi.org/10.4171/jems/ 928

    work page doi:10.4171/jems/ 2019

  12. [12]

    Linear Statistics of Determinantal Point Processes and Norm Rep- resentations,

    M. Levi, J. Marzo, and J. Ortega-Cerdà, “Linear Statistics of Determinantal Point Processes and Norm Rep- resentations,”International Mathematics Research Notices, vol. 2024, no. 19, pp. 12 869–12 903, 2024.DOI: 10.1093/imrn/rnae182

    work page doi:10.1093/imrn/rnae182 2024

  13. [13]

    On the convergence of Riesz means on compact manifolds,

    C. D. Sogge, “On the convergence of Riesz means on compact manifolds,”The Annals of Mathematics, vol. 126, no. 3, p. 439, 1987.DOI:10.2307/1971356

    work page doi:10.2307/1971356 1987

  14. [14]

    Szeg˝o,Orthogonal polynomials(American Mathematical Society Colloquium Publications, V ol

    G. Szeg˝o,Orthogonal polynomials(American Mathematical Society Colloquium Publications, V ol. XXIII), Fourth. American Mathematical Society, Providence, R.I., 1975

    work page 1975

  15. [15]

    Villani,Optimal transport: Old and new

    C. Villani,Optimal transport: Old and new. Springer Berlin Heidelberg, 2009

    work page 2009

  16. [16]

    Two-Point Homogeneous Spaces,

    H.-C. Wang, “Two-Point Homogeneous Spaces,”Annals of Mathematics, vol. 55, no. 1, pp. 177–191, 1952,ISSN: 0003486X. 17

    work page 1952