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arxiv: 2607.00587 · v1 · pith:IWIZO4FPnew · submitted 2026-07-01 · 🧮 math.PR · math-ph· math.MP

(Non-)Hyperuniformity of Second Order Statistics of Point Processes

Fabio Frommer , Martin Hanke This is my paper

Pith reviewed 2026-07-02 07:26 UTC · model grok-4.3

classification 🧮 math.PR math-phmath.MP
keywords point processeshyperuniformitydeterminantal processesGibbs point processesvariance asymptoticssecond order statisticsstructure factorinverse Henderson problem
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The pith

Variance of second-order statistics in both hyperuniform determinantal and non-hyperuniform Gibbs point processes grows proportionally to ball volume.

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

The paper studies the asymptotic variance, as ball radius tends to infinity, of second-order statistics for two families of stationary point processes: determinantal processes whose kernels are projections, and Gibbs processes with superstable pair interactions. It proves that under generic conditions on the kernel or the potential, this variance is asymptotically a positive constant times the volume of the ball. A sympathetic reader cares because first-order hyperuniformity (suppressed fluctuations in point counts) does not automatically extend to second-order measures; the result supplies a concrete counter-example class and an application to the inverse Henderson problem of recovering interactions from pair correlations.

Core claim

Generically, for determinantal point processes with projection kernels and for Gibbs point processes with superstable pair interactions, the variance of the second-order statistics inside a ball centered at the origin is asymptotically proportional to the volume of that ball; the second-order statistics therefore behave non-hyperuniform. The structure factor (or Bartlett spectral measure) of the Gibbs processes is strictly positive, while for the determinantal processes it is positive except for a simple zero at the origin.

What carries the argument

Asymptotic analysis of the variance of second-order statistics, showing volume proportionality under generic conditions on the projection kernel or the superstable interaction potential.

If this is right

  • The structure factor of Gibbs processes with superstable interactions is strictly positive everywhere.
  • For determinantal processes the structure factor vanishes only at the origin and is positive elsewhere.
  • The non-hyperuniform scaling supplies a concrete obstruction for the inverse Henderson problem when the target pair correlation comes from such a process.

Where Pith is reading between the lines

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

  • Hyperuniformity of point counts does not force hyperuniformity of pair-count fluctuations inside the same families.
  • The volume scaling may serve as a diagnostic to distinguish first-order from second-order hyperuniformity in other stationary point processes.

Load-bearing premise

The processes are determinantal with projection kernels or Gibbs with superstable pair interactions, and the kernel or potential satisfies the generic conditions stated in the theorems.

What would settle it

For the Ginibre determinantal process in the plane, compute or simulate the variance of the number of pairs inside a large disk and test whether the growth is strictly linear in area or slower.

read the original abstract

We investigate statistical properties of certain stationary point processes, namely determinantal processes with projection kernels and Gibbs point processes with superstable pair interactions. These are examples of hyperuniform and non-hyperuniform stationary point processes, respectively. We are interested in the variance of their second order statistics within a ball around the origin, and we study the asymptotic growth of this variance as the radius of the ball goes to infinity. It is shown that, generically, for both types of processes the variance is asymptotically proportional to the volume of the ball. In other words: the second order statistics of these point processes behave non-hyperuniform. For Gibbs processes with superstable interactions these results have an interesting application to the so-called inverse Henderson problem of statistical mechanics. We also show that the structure factor (respectively the Bartlett spectral measure) of these Gibbs processes is strictly positive, while it is positive except for a simple zero at the origin for the determinantal processes.

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

1 major / 0 minor

Summary. The paper claims that for stationary determinantal point processes with projection kernels and for Gibbs point processes with superstable pair interactions, the variance of the second-order statistics (within a ball B_R centered at the origin) grows asymptotically proportionally to vol(B_R) as R → ∞, implying non-hyperuniform behavior of these second-order statistics in both classes. It further asserts that the structure factor (Bartlett spectral measure) is strictly positive for the Gibbs case and positive except for a simple zero at the origin for the determinantal case, with an application to the inverse Henderson problem.

Significance. If the central claims on variance asymptotics hold, the work would provide a clear separation between hyperuniformity of first-order versus second-order statistics and furnish a concrete application to statistical mechanics via the inverse Henderson problem. The positivity results for the spectral measures would also be of independent interest in the theory of point processes.

major comments (1)
  1. [Abstract] Abstract: the claim that the variance of the second-order statistics inside B_R is asymptotically proportional to vol(B_R) for determinantal projection-kernel processes is in tension with the stated property that the Bartlett spectral measure has a simple zero at the origin. In the Fourier representation Var = ∫ S(k) |χ̂_R(k)|^2 dk, a simple zero S(k) ∼ |k| near k=0 produces, after the change of variables k = u/R, an extra factor R^{-1} and therefore growth of order vol(B_R)/R rather than vol(B_R). This scaling issue is load-bearing for the central non-hyperuniformity claim that is asserted for both classes.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their detailed reading and for highlighting the scaling tension in the abstract. We address the comment below and will make the necessary revisions.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the claim that the variance of the second-order statistics inside B_R is asymptotically proportional to vol(B_R) for determinantal projection-kernel processes is in tension with the stated property that the Bartlett spectral measure has a simple zero at the origin. In the Fourier representation Var = ∫ S(k) |χ̂_R(k)|^2 dk, a simple zero S(k) ∼ |k| near k=0 produces, after the change of variables k = u/R, an extra factor R^{-1} and therefore growth of order vol(B_R)/R rather than vol(B_R). This scaling issue is load-bearing for the central non-hyperuniformity claim that is asserted for both classes.

    Authors: We agree with the referee's Fourier analysis. The simple zero S(k) ∼ |k| at the origin does indeed produce the extra R^{-1} factor under the change of variables, yielding asymptotic growth of order vol(B_R)/R for the determinantal processes rather than vol(B_R). The abstract statement that the variance is asymptotically proportional to vol(B_R) for both classes is therefore imprecise for the determinantal case. We will revise the abstract, the introduction, and the relevant theorem statements to distinguish the two classes: for Gibbs processes with superstable interactions the variance grows proportionally to vol(B_R), while for determinantal projection-kernel processes it grows as vol(B_R)/R. We will also adjust the non-hyperuniformity phrasing to reflect this distinction accurately while preserving the application to the inverse Henderson problem for the Gibbs case. revision: yes

Circularity Check

0 steps flagged

No circularity detected; derivations rely on independent properties of point process classes

full rationale

The paper establishes asymptotic variance results for second-order statistics of determinantal projection-kernel processes and Gibbs processes with superstable pair interactions by invoking standard spectral properties (Bartlett measure with simple zero at origin for the former; strict positivity for the latter) and generic conditions on kernels/interactions. These steps draw on established theory of the respective process classes without any reduction of the target variance quantity to a fitted parameter, self-defined input, or load-bearing self-citation chain. No equation or theorem in the provided material equates the claimed ~vol scaling to its own definition or to a prior result by the same authors that itself assumes the conclusion. The inverse Henderson application is presented as a consequence rather than a definitional loop.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

The central claims rest on the stationarity of the processes, the projection-kernel property for determinantal processes, and superstable pair interactions for Gibbs processes; these are standard domain assumptions rather than new free parameters or invented entities.

axioms (3)
  • domain assumption The point processes are stationary.
    Invoked throughout the abstract to define the second-order statistics and their variance inside balls centered at the origin.
  • domain assumption Determinantal processes use projection kernels.
    Stated as the class that yields hyperuniform examples.
  • domain assumption Gibbs processes have superstable pair interactions.
    Stated as the class that yields non-hyperuniform examples and the inverse Henderson application.

pith-pipeline@v0.9.1-grok · 5692 in / 1431 out tokens · 17220 ms · 2026-07-02T07:26:13.256291+00:00 · methodology

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Reference graph

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