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

Carlos Beltr\'an; Giacomo Gigante; Pedro R. L\'opez-G\'omez; Ryan W. Matzke; Uju\'e Etayo

arxiv: 2605.22295 · v1 · pith:TJZXETTEnew · submitted 2026-05-21 · 🧮 math.CA · math.PR

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

Carlos Beltr\'an , Uju\'e Etayo , Giacomo Gigante , Pedro R. L\'opez-G\'omez , Ryan W. Matzke This is my paper

Pith reviewed 2026-05-22 02:07 UTC · model grok-4.3

classification 🧮 math.CA math.PR

keywords determinantal point processesdiscrepancytwo-point homogeneous spacesharmonic ensembleprojective ensemblecompact manifoldsmetric balls

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The pith

Determinantal point processes on spheres and projective spaces achieve discrepancy bounds scaling as the square root of N to the power 1 minus 1 over D times log factors.

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

The paper derives upper bounds on the L-infinity discrepancy of point sets sampled from determinantal point processes on all compact connected two-point homogeneous spaces, which include spheres and projective spaces. It relies on concentration inequalities together with variance estimates for the number of points falling inside metric balls to produce general upper bounds that apply to any homogeneous DPP. For the harmonic ensemble the discrepancy of N points is O of the square root of N to the power 1-1/D times log N with high probability. For the projective ensemble on complex projective space the bound tightens to the square root of N to the power 1-1/D times log N. These estimates extend earlier results that had been available only for the sphere.

Core claim

Using concentration inequalities and variance estimates for the number of points in metric balls that hold uniformly on all compact connected two-point homogeneous spaces, the paper shows that for the harmonic ensemble the discrepancy of N points is O((N^{1-1/D})^{1/2} log N) with high probability, while for the projective ensemble on CP^d it is the sharper O((N^{1-1/D} log N)^{1/2}).

What carries the argument

Concentration inequalities and variance estimates for the number of points in metric balls, used uniformly across the spaces to control the L-infinity discrepancy of homogeneous determinantal point processes.

If this is right

The same discrepancy upper bounds apply to any homogeneous determinantal point process on these spaces.
The projective ensemble on CP^d improves the rate by keeping the log factor inside the square root.
The results extend all previously known discrepancy controls from the sphere alone to the full class of compact connected two-point homogeneous spaces.
Such controlled discrepancy supports more accurate numerical integration or sampling on these manifolds.

Where Pith is reading between the lines

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

If the uniform variance and concentration estimates are robust, analogous discrepancy controls may hold for other kernel-based sampling schemes on the same manifolds.
The difference in rates between harmonic and projective ensembles indicates that kernel choice can tighten the log factor without changing the main scaling.
Direct numerical checks of the discrepancy for moderate N on the sphere or on CP^1 would test whether the log factor is sharp or can be removed.

Load-bearing premise

The concentration inequalities and variance estimates for the number of points in metric balls must hold uniformly over all radii and all compact connected two-point homogeneous spaces.

What would settle it

Generating many independent harmonic-ensemble samples of N points on the two-sphere and observing that their discrepancy exceeds C times sqrt(N^{1/2}) log N for some large constant C and large N would contradict the claimed bound.

Figures

Figures reproduced from arXiv: 2605.22295 by Carlos Beltr\'an, Giacomo Gigante, Pedro R. L\'opez-G\'omez, Ryan W. Matzke, Uju\'e Etayo.

read the original 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.

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.

Desk Editor's Note private letter to a colleague

Extends DPP discrepancy bounds from spheres to all compact connected two-point homogeneous spaces, with a modestly sharper rate for the projective ensemble on CP^d.

read the letter

The main takeaway is that this paper carries discrepancy bounds for determinantal point processes from the sphere over to the full list of compact connected two-point homogeneous spaces and tightens the rate a bit for the projective ensemble on complex projective space. They obtain O((N^{1-1/D})^{1/2} log N) with high probability for the harmonic ensemble across these manifolds and O((N^{1-1/D} log N)^{1/2}) for the projective case on CP^d. The work uses concentration inequalities on the number of points in metric balls together with variance estimates coming from the reproducing kernel, then integrates to control the sup-norm discrepancy. Two-point homogeneity is what lets them treat the kernels uniformly enough to get the same order as the earlier sphere results. That step is not automatic, because the geometry changes with the space, so the authors had to adjust the variance calculations accordingly. The derivations rest on standard tools rather than new concentration machinery, which keeps the argument clean. The soft spot worth checking is whether the variance estimates for ball counts really stay uniform without picking up manifold-specific factors. The stress-test note flags this exact issue: curvature or dimension effects could introduce constants that the O notation swallows but that would matter if you wanted explicit dependence on the space. If the proofs show homogeneity absorbs everything, the bounds hold as stated; otherwise the general upper-bound section needs a closer look. A reader already working on discrepancy or point processes on symmetric spaces will find this useful for filling in the picture on these special manifolds. It does not claim results for arbitrary spaces, so it stays within its niche. The math is grounded in existing literature and the claims are falsifiable, so the paper deserves a serious referee who can verify the variance uniformity. I would send it to peer review.

Referee Report

1 major / 2 minor

Summary. The manuscript studies the L^∞ discrepancy of determinantal point processes (DPPs) on compact connected two-point homogeneous spaces (spheres, RP^d, CP^d, HP^d, OP^2). It derives general upper bounds for homogeneous DPPs via concentration inequalities and variance estimates on the number of points in metric balls, then specializes to the harmonic ensemble (yielding O((N^{1-1/D})^{1/2} log N) with high probability, D the real dimension) and to the projective ensemble on CP^d (sharper O((N^{1-1/D} log N)^{1/2})). The results extend prior sphere-only bounds.

Significance. If the uniformity claims hold, the work provides the first discrepancy rates for DPPs on the full list of compact two-point homogeneous spaces, strengthening the link between DPP sampling and low-discrepancy sets on symmetric manifolds. The explicit rates and the distinction between harmonic and projective ensembles are useful for applications in numerical integration and quasi-Monte Carlo methods on non-Euclidean domains.

major comments (1)

[General upper bounds section] General upper bounds section: the claim that variance estimates for the number of points in metric balls are uniform across all compact connected two-point homogeneous spaces (with constants independent of the specific manifold) is invoked to obtain the stated O((N^{1-1/D})^{1/2} log N) rate, but the derivation of the variance (via squared-kernel integrals over balls) is not shown to absorb curvature or dimension-dependent multipliers uniformly; a concrete check that the constants remain O(1) when passing from S^D to RP^d, CP^d, etc., is needed to support the general bound.

minor comments (2)

The notation for the reproducing kernels and the precise definition of the harmonic versus projective ensembles should be recalled explicitly in the statement of the main theorems rather than only in the preliminaries.
A short table comparing the new rates with the earlier sphere results would improve readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments on the manuscript. We address the single major comment below and will incorporate the requested clarification in the revised version.

read point-by-point responses

Referee: General upper bounds section: the claim that variance estimates for the number of points in metric balls are uniform across all compact connected two-point homogeneous spaces (with constants independent of the specific manifold) is invoked to obtain the stated O((N^{1-1/D})^{1/2} log N) rate, but the derivation of the variance (via squared-kernel integrals over balls) is not shown to absorb curvature or dimension-dependent multipliers uniformly; a concrete check that the constants remain O(1) when passing from S^D to RP^d, CP^d, etc., is needed to support the general bound.

Authors: We thank the referee for this observation. In the general upper bounds, the variance of the number of points in a metric ball B is controlled by the double integral of |K(x,y)|^2 over B×B, where K is the reproducing kernel of the underlying Hilbert space of functions. Because the DPP is homogeneous on a two-point homogeneous space, K depends only on geodesic distance, and the integral reduces to a one-dimensional radial integral against the standard volume density of the space. These volume densities are explicit trigonometric polynomials whose coefficients depend only on the real dimension D (e.g., sin^{D-1} r on spheres, sin^{2d-1} r cos r on CP^d, etc.). On the compact interval [0,π] all such densities are bounded above and below by constants that depend solely on D and not on the particular space beyond its dimension. Consequently the variance bound carries a multiplicative factor that is O(1) uniformly across the classified list of spaces once D is fixed. We agree, however, that this uniformity was left implicit. In the revision we will insert a short lemma (or a dedicated paragraph in Section 3) that records the explicit comparison of the volume forms and verifies that the resulting constant is indeed independent of the concrete manifold. revision: yes

Circularity Check

0 steps flagged

Derivations rely on external concentration inequalities; no reduction of claims to self-fitted inputs or load-bearing self-citations

full rationale

The paper's central bounds are obtained by applying concentration inequalities and variance estimates for the number of points in metric balls to control the L^∞ discrepancy, then extending these via the reproducing kernel and two-point homogeneity properties to all listed spaces. These estimates are invoked as holding uniformly (general upper bounds section) without the final O-notation being forced by a parameter fit inside the paper or by a self-citation chain that itself reduces to the target result. Prior sphere results are extended rather than presupposed as the sole justification, leaving independent mathematical content in the uniformity argument and the specific rates for the harmonic and projective ensembles.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review; no free parameters, ad-hoc axioms, or invented entities are visible. The work rests on standard properties of DPPs and known concentration tools for homogeneous spaces.

pith-pipeline@v0.9.0 · 5704 in / 1118 out tokens · 33630 ms · 2026-05-22T02:07:27.189070+00:00 · methodology

discussion (0)

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Relation between the paper passage and the cited Recognition theorem.

We study the L^∞ discrepancy of point sets generated by determinantal point processes on all compact, connected two-point homogeneous spaces... For the harmonic ensemble, 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.

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matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

13 extracted references · 13 canonical work pages

[1]

Alishahi and M

K. Alishahi and M. Zamani,The spherical ensemble and uniform distribution of points on the sphere, Electronic Journal of Probability20(2015), no. 23, 27.↑3, 5

work page 2015
[2]

Anderson, M

A. Anderson, M. Dostert, P. J. Grabner, R. W. Matzke, and T. A. Stepaniuk,Riesz and Green energy on projective spaces, Transactions of the American Mathematical Society. Series B10(2023), 1039–1076.↑2, 4, 11

work page 2023
[3]

Beck,Some upper bounds in the theory of irregularities of distribution, Acta Arithmetica 43(1984), no

J. Beck,Some upper bounds in the theory of irregularities of distribution, Acta Arithmetica 43(1984), no. 2, 115–130.↑2, 8, 9

work page 1984
[4]

Beck and W

J. Beck and W. W. L. Chen,Irregularities of distribution, Cambridge Tracts in Mathematics, vol. 89, Cambridge University Press, Cambridge, 1987.↑2

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[5]

Beltrán and U

C. Beltrán and U. Etayo,The projective ensemble and distribution of points in odd-dimensional spheres, Constructive Approximation48(2018), no. 1, 163–182.↑5

work page 2018
[6]

Beltrán, J

C. Beltrán, J. Marzo, and J. Ortega-Cerdà,Energy and discrepancy of rotationally invariant determinantal point processes in high dimensional spheres, Journal of Complexity37(2016), 76–109.↑2, 3, 4, 10

work page 2016
[7]

Borda, P

B. Borda, P. Grabner, and R. W. Matzke,Riesz energy,L2 discrepancy, and optimal transport of determinantal point processes on the sphere and the flat torus, Mathematika70(2024), no. 2, e12245.↑3

work page 2024
[8]

Brandolini, W

L. Brandolini, W. W. L. Chen, L. Colzani, G. Gigante, and G. Travaglini,Discrepancy and numerical integration on metric measure spaces, Journal of Geometric Analysis29(2019), no. 1, 328–369.↑2

work page 2019
[9]

Brandolini, B

L. Brandolini, B. Gariboldi, and G. Gigante,Irregularities of distribution on two-point homogeneous spaces, The mathematical heritage of Guido Weiss, 2025, pp. 101–125.↑11

work page 2025
[10]

J. B. Hough, M. Krishnapur, Y. Peres, and B. Virág,Zeros of Gaussian analytic functions and determinantal point processes, University Lecture Series, vol. 51, American Mathematical Society, Providence, RI, 2009.↑3, 4, 8

work page 2009
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Rider and B

B. Rider and B. Virág,Complex determinantal processes andH 1 noise, Electronic Journal of Probability12(2007), no. 45, 1238–1257.↑10

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[12]

M. M. Skriganov,Point distributions in two-point homogeneous spaces, Mathematika65 (2019), no. 3, 557–587.↑2 16 C. BELTRÁN, U. ETAYO, G. GIGANTE, P. R. LÓPEZ-GÓMEZ, AND R. W. MATZKE

work page 2019
[13]

G. Szegö,Orthogonal polynomials, 4th ed., American Mathematical Society, 1975.↑12 Carlos Beltrán: Departamento de Matemáticas, Estadística y Computación, Universi- dad de Cantabria, A vda. Los Castros, s/n, 39005 Santander, Spain Email address:beltranc@unican.es Ujué Etayo: Departamento de Matemáticas, CUNEF Universidad, C/ Almansa, 101, 28040 Madrid, Spa...

work page 1975

[1] [1]

Alishahi and M

K. Alishahi and M. Zamani,The spherical ensemble and uniform distribution of points on the sphere, Electronic Journal of Probability20(2015), no. 23, 27.↑3, 5

work page 2015

[2] [2]

Anderson, M

A. Anderson, M. Dostert, P. J. Grabner, R. W. Matzke, and T. A. Stepaniuk,Riesz and Green energy on projective spaces, Transactions of the American Mathematical Society. Series B10(2023), 1039–1076.↑2, 4, 11

work page 2023

[3] [3]

Beck,Some upper bounds in the theory of irregularities of distribution, Acta Arithmetica 43(1984), no

J. Beck,Some upper bounds in the theory of irregularities of distribution, Acta Arithmetica 43(1984), no. 2, 115–130.↑2, 8, 9

work page 1984

[4] [4]

Beck and W

J. Beck and W. W. L. Chen,Irregularities of distribution, Cambridge Tracts in Mathematics, vol. 89, Cambridge University Press, Cambridge, 1987.↑2

work page 1987

[5] [5]

Beltrán and U

C. Beltrán and U. Etayo,The projective ensemble and distribution of points in odd-dimensional spheres, Constructive Approximation48(2018), no. 1, 163–182.↑5

work page 2018

[6] [6]

Beltrán, J

C. Beltrán, J. Marzo, and J. Ortega-Cerdà,Energy and discrepancy of rotationally invariant determinantal point processes in high dimensional spheres, Journal of Complexity37(2016), 76–109.↑2, 3, 4, 10

work page 2016

[7] [7]

Borda, P

B. Borda, P. Grabner, and R. W. Matzke,Riesz energy,L2 discrepancy, and optimal transport of determinantal point processes on the sphere and the flat torus, Mathematika70(2024), no. 2, e12245.↑3

work page 2024

[8] [8]

Brandolini, W

L. Brandolini, W. W. L. Chen, L. Colzani, G. Gigante, and G. Travaglini,Discrepancy and numerical integration on metric measure spaces, Journal of Geometric Analysis29(2019), no. 1, 328–369.↑2

work page 2019

[9] [9]

Brandolini, B

L. Brandolini, B. Gariboldi, and G. Gigante,Irregularities of distribution on two-point homogeneous spaces, The mathematical heritage of Guido Weiss, 2025, pp. 101–125.↑11

work page 2025

[10] [10]

J. B. Hough, M. Krishnapur, Y. Peres, and B. Virág,Zeros of Gaussian analytic functions and determinantal point processes, University Lecture Series, vol. 51, American Mathematical Society, Providence, RI, 2009.↑3, 4, 8

work page 2009

[11] [11]

Rider and B

B. Rider and B. Virág,Complex determinantal processes andH 1 noise, Electronic Journal of Probability12(2007), no. 45, 1238–1257.↑10

work page 2007

[12] [12]

M. M. Skriganov,Point distributions in two-point homogeneous spaces, Mathematika65 (2019), no. 3, 557–587.↑2 16 C. BELTRÁN, U. ETAYO, G. GIGANTE, P. R. LÓPEZ-GÓMEZ, AND R. W. MATZKE

work page 2019

[13] [13]

G. Szegö,Orthogonal polynomials, 4th ed., American Mathematical Society, 1975.↑12 Carlos Beltrán: Departamento de Matemáticas, Estadística y Computación, Universi- dad de Cantabria, A vda. Los Castros, s/n, 39005 Santander, Spain Email address:beltranc@unican.es Ujué Etayo: Departamento de Matemáticas, CUNEF Universidad, C/ Almansa, 101, 28040 Madrid, Spa...

work page 1975