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arxiv: 2605.22803 · v1 · pith:MA23BESAnew · submitted 2026-05-21 · 🧮 math.PR · cond-mat.dis-nn· cond-mat.soft

Persistence of asymptotic variance under transport: from hyperfluctuation to stealthy hyperuniformity

Luca Lotz , Michael A. Klatt This is my paper

Pith reviewed 2026-05-22 02:55 UTC · model grok-4.3

classification 🧮 math.PR cond-mat.dis-nncond-mat.soft
keywords p-uniformityhyperuniformitypoint processestransport mapsdensity fluctuationsasymptotic variancehyperfluctuationstochastic geometry
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The pith

Transport preserves p-uniformity of density fluctuations when the map has finite (d+p)th moment.

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

The paper introduces p-uniformity to characterize the scaling of density fluctuations in spatial random systems in any dimension, ranging from hyperfluctuation to stealthy hyperuniformity. Its central theorem gives sufficient conditions under which this property is preserved when points are moved according to a transport map. The conditions require a finite (d+p)th moment of the transport distance to support a Taylor expansion, plus a bound on the resulting remainder terms. This resolves and broadens a prior open problem to cover arbitrary p-uniform sources, any dimension, and cases where the source depends on the transport. The theorem directly yields constructions of new isotropic point processes that achieve arbitrarily high p-uniformity and admit linear-time simulation.

Core claim

Our central theorem establishes sufficient conditions to preserve p-uniformity under transport. The first condition, a finite (d+p)-th moment of the transport distance, allows for a Taylor expansion of the transport. The second condition controls the corresponding terms. We thus solve a previously stated open problem; indeed we extend it, since our result applies to a general p-uniform source in any dimension, and the source and transport may be dependent. As an application, we construct new classes of point processes that are isotropic and p-uniform with arbitrarily high p, and that can be simulated in linear time.

What carries the argument

p-uniformity, the scaling property of density fluctuations that interpolates between hyperfluctuation and stealthy hyperuniformity

Load-bearing premise

The transport distance satisfies a finite (d+p)th moment condition that permits a controlled Taylor expansion of the fluctuation scaling, together with a bound on the remainder terms.

What would settle it

A concrete transport map whose (d+p)th moment is infinite and under which the scaling exponent of density fluctuations changes would falsify preservation of p-uniformity.

Figures

Figures reproduced from arXiv: 2605.22803 by Luca Lotz, Michael A. Klatt.

Figure 1
Figure 1. Figure 1: Illustration of our construction of an isotropic point process that is hyperuniform [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
read the original abstract

We introduce $p$-uniformity to characterize the scaling of density fluctuations in spatial random systems in $\mathbb{R}^d$, ranging from hyperfluctuation to stealthy hyperuniformity. Our central theorem establishes sufficient conditions to preserve $p$-uniformity under transport. The first condition, a finite $(d+p)$-th moment of the transport distance, allows for a Taylor expansion of the transport. The second condition controls the corresponding terms. We thus solve a previously stated open problem; indeed we extend it, since our result applies to a general $p$-uniform source in any dimension, and the source and transport may be dependent. As an application, we construct new classes of point processes that are isotropic and $p$-uniform with arbitrarily high $p$, and that can be simulated in linear time. We conclude with an outlook on a converse statement.

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

2 major / 3 minor

Summary. The paper introduces p-uniformity to characterize the scaling of density fluctuations in spatial random systems in R^d, ranging from hyperfluctuation to stealthy hyperuniformity. Its central theorem establishes sufficient conditions to preserve p-uniformity under transport: a finite (d+p)-th moment of the transport distance allowing a Taylor expansion, and a second condition controlling the terms. This solves a previously stated open problem, extending it to general p-uniform sources in any dimension with possible dependence between source and transport. Applications include new isotropic p-uniform point processes with high p simulatable in linear time, and an outlook on a converse.

Significance. If the central theorem holds, the result is significant for providing a general mechanism to transfer fluctuation scaling via transport in point processes, solving an open problem while extending to dependent cases and arbitrary dimensions. The constructions of isotropic high-p examples and the linear-time simulation method are practical strengths that enable reproducible examples and falsifiable variance predictions. Credit is due for the direct, assumption-light extension without requiring independence.

major comments (2)
  1. [§3] §3 (central theorem): the second condition controlling remainder terms after the Taylor expansion of the fluctuation integral must be stated as an explicit bound (e.g., an inequality on the transport map) to confirm it yields the required o(r^{-p}) error without hidden uniformity assumptions; this is load-bearing for the preservation claim.
  2. [§4] §4 (applications): the linear-time simulation claim for the constructed processes requires an explicit complexity analysis showing that applying the transport map does not exceed linear cost in the number of points, given the base process.
minor comments (3)
  1. [Abstract] Abstract: briefly define or reference 'stealthy hyperuniformity' for accessibility, as the term is used without prior introduction.
  2. [Introduction] Introduction: cite the specific prior work stating the open problem being solved, rather than referring to it generically.
  3. [Notation] Notation throughout: ensure the definition of p-uniformity (the precise o(r^{-p}) variance scaling) is restated or cross-referenced when used in the theorem and applications.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their positive evaluation of our manuscript and for the constructive major comments. We address each point below and indicate the planned revisions.

read point-by-point responses

  1. Referee: [§3] §3 (central theorem): the second condition controlling remainder terms after the Taylor expansion of the fluctuation integral must be stated as an explicit bound (e.g., an inequality on the transport map) to confirm it yields the required o(r^{-p}) error without hidden uniformity assumptions; this is load-bearing for the preservation claim.

    Authors: We agree that the second condition should be stated explicitly to ensure the o(r^{-p}) error is controlled without implicit uniformity assumptions. In the revised manuscript we will reformulate this condition as a concrete inequality involving the transport map and the source measure, followed by a direct verification that the resulting remainder is indeed o(r^{-p}). revision: yes

  2. Referee: [§4] §4 (applications): the linear-time simulation claim for the constructed processes requires an explicit complexity analysis showing that applying the transport map does not exceed linear cost in the number of points, given the base process.

    Authors: We thank the referee for this observation. The transport map is a fixed, pointwise function whose evaluation cost is independent of the number of points. We will add an explicit complexity paragraph in Section 4 confirming that, when the base process is generated in linear time, the overall procedure remains O(N) for N points. revision: yes

Circularity Check

0 steps flagged

No significant circularity; theorem rests on explicit moment conditions and Taylor expansion

full rationale

The central result is a theorem giving sufficient conditions (finite (d+p)-th moment on transport distance plus remainder bound) that justify a controlled Taylor expansion preserving p-uniformity. This is a direct analytic argument on the fluctuation scaling integral, with no reduction of the conclusion to a self-definition, a fitted parameter renamed as prediction, or a load-bearing self-citation chain. The extension to general p-uniform sources and dependent transport follows from the stated hypotheses rather than from prior author work being invoked as an unverified uniqueness theorem. The construction of high-p isotropic examples is an application of the theorem, not a renaming of known patterns. The derivation is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The paper introduces the new concept of p-uniformity and relies on standard moment assumptions from probability theory plus the control condition on Taylor remainders; no free parameters or data-fitting steps are mentioned.

axioms (2)
  • domain assumption The source point process is p-uniform in R^d
    This is the starting point for the preservation theorem stated in the abstract.
  • domain assumption The transport distance possesses a finite (d+p)-th moment
    Invoked to justify the Taylor expansion of the transport map.
invented entities (1)
  • p-uniformity no independent evidence
    purpose: A parameterized characterization of density-fluctuation scaling ranging from hyperfluctuation to stealthy hyperuniformity
    New definition introduced to unify and extend existing notions of hyperuniformity.

pith-pipeline@v0.9.0 · 5684 in / 1490 out tokens · 57159 ms · 2026-05-22T02:55:25.692998+00:00 · methodology

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