A genuine test for hyperuniformity Michael A. Klatt G unter Lastand Norbert Henze October 25 2022

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A genuine test for hyperuniformity

Michael A. Klatt∗

, G¨unter Last†and Norbert Henze‡

October 25, 2022

Abstract

We devise the ﬁrst rigorous signiﬁcance test for hyperuniformity with sensitive re-

sults, even for a single sample. Our starting point is a detailed study of the empirical

Fourier transform of a stationary point process on Rd. For large system sizes, we

derive the asymptotic covariances and prove a multivariate central limit theorem

(CLT). The scattering intensity is then used as the standard estimator of the struc-

ture factor. The above CLT holds for a preferably large class of point processes,

and whenever this is the case, the scattering intensity satisﬁes a multivariate limit

theorem as well. Hence, we can use the likelihood ratio principle to test for hyper-

uniformity. Remarkably, the asymptotic distribution of the resulting test statistic is

universal under the null hypothesis of hyperuniformity. We obtain its explicit form

from simulations with very high accuracy. The novel test precisely keeps a nominal

signiﬁcance level for hyperuniform models, and it rejects non-hyperuniform exam-

ples with high power even in borderline cases. Moreover, it does so given only a

single sample with a practically relevant system size.

Keywords: point process, hyperuniformity, structure factor, scattering intensity, central

limit theorem, likelihood ratio test

2020 Mathematics Subject Classiﬁcation: 62H11; 60G55

1 Introduction

Disordered hyperuniform point patterns are characterized by an anomalous suppression

of density ﬂuctuations on large scales [39]. They can be both isotropic like a liquid and

homogeneous like a crystal. In that sense, they represent a new state of matter, and

they have attracted a quickly growing attention in physics [41, 13, 39, 35, 31, 43, 37],

biology [26, 25], material science [9, 45, 16], and probability theory [14, 15, 32, 8, 33], to

name just a few examples. What has, however, been missing so far is a statistical test

∗klattm@hhu.de; Institut f¨ur Theoretische Physik II: Weiche Materie, Heinrich-Heine-Universit¨at

D¨usseldorf, 40225 D¨usseldorf, Germany; Experimental Physics, Saarland University, Center for Bio-

physics, 66123 Saarbr¨ucken, Germany.

†guenter.last@kit.edu; Karlsruhe Institute of Technology, Institute for Stochastics, 76131 Karlsruhe,

Germany.

‡norbert.henze@kit.edu; Karlsruhe Institute of Technology, Institute for Stochastics, 76131 Karlsruhe,

Germany.

arXiv:2210.12790v1 [math.ST] 23 Oct 2022

that, for a given point pattern, can reliably assess the hypothesis of hyperuniformity. This

lack inevitably entails the risk of ambiguous ﬁndings.

Hyperuniformity of a stationary point process is a second order property, and it can

be deﬁned in either real or Fourier space. In the former case, a point process is said

to be hyperuniform if, in the limit of large window sizes, the variance of the number of

points in an observation window grows more slowly than the volume of the window. An

alternative deﬁnition is that the so-called structure factor (see [18] and (2.8)) vanishes

for small wave vectors. The structure factor is essentially the Fourier transform of the

reduced pair correlation function, which is a key concept of the theory of point processes

[10, 34].

Both of these deﬁnitions allow for ad hoc tests of hyperuniformity based on heuristic

principles, and such tests have so far been used in the applied literature. A straightforward

approach is to estimate the variance of the number of points as a function of the window

size, see, e.g., [12, 37]. However, such a real space sampling requires heuristic rules that

avoid a signiﬁcant overlap between sampling windows [29, 40]. While a rigorous test of

hyperuniformity could be deﬁned using independent samples for each observation window,

such a procedure would require millions of samples [40], which is unrealistic for practical

applications.

An alternative approach is to extrapolate the structure factor in the limit of small wave

vectors [12, 1, 31]. The usual heuristic approach consists in using a least-squares ﬁt that

does not account for the non-negativity of the structure factor and hence does not result

in a rigorous signiﬁcance test. This inconsistency is avoided in a recent preprint [23] that

provides a detailed comparison of diﬀerent estimators of the structure factor. However,

similar to the real space sampling outlined above, also this hypothesis test comes at the

price of requiring an independent sample for each data point.

In this paper, we overcome the limitations of previous ad hoc methods by exploiting

detailed distributional properties of the empirical Fourier transform. More speciﬁcally,

we consider point processes whose estimated structure factor satisﬁes a multivariate limit

theorem. We expect that this limit theorem holds for a preferably large class of point

processes. This assumption is corroborated by our Theorem 4.1 and by [4, Theorem 1].

Moreover, we support our hypothesis by extensive simulations of several point processes.

By working in an asymptotic setting, we introduce a rigorous likelihood ratio test that

sensitively distinguishes a hyperuniform from a non-hyperuniform sample for practically

relevant models and parameters. The necessary approximations hold in the limit of large

sample sizes, which is a natural requirement for the study of a long-range property such

as hyperuniformity. Notably, high-precision simulations demonstrate that these approxi-

mations are accurate even for a moderate system size of about a 1000 points per sample.

To highlight how diﬃcult it can be to distinguish a hyperuniform point pattern from

a non-hyperuniform one by eye, Figure 1 is illustrative. The left hand ﬁgure shows a

realization of a (non-hyperuniform) Mat´ern III process close to saturation [10]. While this

process exhibits a high degree of local order, it possesses Poisson-like long-range density

ﬂuctuations that are incompatible with hyperuniformity. In juxtaposition, the right hand

ﬁgure illustrates a hyperuniform process. To construct this sample, we ﬁrst simulated a

stealthy hyperuniform point pattern [39] and then perturbed each point — independently

of each other — according to a Gaussian distribution. Both processes have been simulated

on the ﬂat torus. The subtle diﬀerence in the long-range order of the two samples is clearly

Figure 1: Realizations of a non-hyperuniform (left) and a hyperuniform (right) point

process, both with about 10000 points.

detected by the novel test statistic. This statistic takes a value larger than 300 for the non-

hyperuniform sample, i.e., it strongly exceeds the critical value of 2.39, which corresponds

to a nominal signiﬁcance level of 5%. In contrast, the test statistic attains the value 0.04

for the hyperuniform sample. This fact clearly demonstrates a striking sensitivity of our

test for hyperuniformity.

In the following, we summarize our results and relate them to the relevant literature. In

Section 2, we introduce a few key concepts from the theory of stationary point processes,

and we deﬁne hyperuniformity. Following the classical reference [6] (see also the recent

survey [23]), Section 3 introduces the empirical Fourier transform of a point process, de-

ﬁned in terms of a rather general taper function. Proposition 3.3 provides the asymptotic

covariance structure of this transform. At least implicitly, the special case d= 1 has been

dealt with in [6]. Even though this structure might be considered part of the folklore in

spatial statistics, we are not aware of a rigorously proved result for a general dimension.

A related paper is [19], where the authors study bias and variance of kernel estimators of

the asymptotic variance. In Section 4, we establish a multivariate central limit theorem

(CLT) for the empirical Fourier transform (see Theorem 4.1), thus extending a result of

[6] to general dimensions. And as in [6] we do so under an assumption on the reduced

factorial cumulant measures. Our proof does also beneﬁt from the classical source [22],

where the authors adopt a similar hypothesis to establish a related CLT; see also [21] for

a more recent contribution to the asymptotic normality of kernel estimators of correlation

functions. We illustrate our result with Poisson cluster and α-determinantal processes.

In what follows, we use the empirical scattering intensity (2.13) as a well-established

estimator of the structure factor; see [39]. Section 5 discusses its multivariate asymp-

totic behavior for diﬀerent wave vectors and for large system sizes. It will be seen that

the resulting limit distribution is made up of exponential random variables which are

independent for diﬀerent wave vectors (Proposition 5.4). This crucial result requires the

underlying point process to satisfy the multivariate central limit theorem (5.2). On the

theoretical side, this assumption is supported by our Theorem 4.1 and by [4, Theorem

1]; see Remark 5.3. Moreover, we have simulated six models (two of which are hyperuni-

form) across the ﬁrst three dimensions. In each of these cases we observed the theoretically

predicted behavior over up to six orders of magnitude already for small system sizes of

about 1000 points. Hence, our simulations indicate a fast speed of convergence, so that

the asymptotic distribution is attained with high accuracy at practically relevant system

sizes.

In Section 6, we abandon the background of point processes and adopt the asymptotic

setting of Proposition 5.4. By taking a quadratic parametrization (2.9) of the structure

factor, we obtain a statistical model with only two parameters sand t(say). This addi-

tional approximation and simpliﬁcation can be justiﬁed by taking suﬃciently large system

sizes. Hyperuniformity is then characterized by the equation s= 0, which deﬁnes our

null hypothesis H0. Next we introduce maximum likelihood estimators (MLEs) b

t0of t

under the null hypothesis and (bs, b

t1) of (s, t) for the full model, respectively. Under H0,

the estimator b

t0of t, when applied to random data, is unbiased and consistent. For ﬁnite

system sizes, it follows an exact gamma distribution. Under the full model we observe

that, asymptotically, the distribution of bsequals a mixture of the Dirac distribution in

0 and a gamma distribution, while b

t1follows a gamma distribution. In particular the

distribution of bshas an atom in 0. These ﬁndings are based on numerical solutions for

the likelihood equations and on extensive simulations.

In Section 7, we introduce our likelihood ratio test. Working in the asymptotic and

parametric setting of Section 6, we consider the ratio of the maximum likelihood of the

data under the hypothesis and the corresponding maximum likelihood over the full pa-

rameter space. The resulting test statistic is twice the negative logarithm of this ratio.

Analytically, the asymptotic distribution of this test statistic seems to be out of reach.

Again we have run extensive simulations to obtain this distribution with high accuracy.

Under the hypothesis, this limit distribution is a mixture of an atom at 0 (inherited from

the atom of bs) and a gamma distribution, and it turns out that it does already apply for

rather moderate system sizes. Moreover, we show rigorously that this limit distribution

does not depend on the actual value of the parameter t. Therefore we obtain a signiﬁ-

cance test for hyperuniformity which can conveniently be used to spatial point patterns

occurring in applications.

In Section 8 we apply our test to independent thinnings of the matching process from

[32]; see also Example (v) in Section 5. The introduction of a thinning parameter is a

convenient way of tuning the parameter s. The hyperuniform case s= 0 arises in the

limit case of no thinning. It turns out that the asymptotic distribution of the test statistic

applies extremely well to this point process. Remarkably, it does so for only one sample

with moderately large system size. We apply our test with a nominal signiﬁcance level of

0.05 for diﬀerent values of the parameter s. If s= 0 the test keeps this nominal level very

well. Already for s= 10−3and a system size of L= 150 the test rejects hyperuniformity

in almost all cases. Even for s= 10−4, which is an order of magnitude smaller than

typical non-hyperuniform models (see [39, 31]), the rejection rate increases dramatically

with growing systems size; see Table 1 for more details.

We ﬁnish the paper with some remarks on the principles underlying our test, and we

also formulate some interesting further problems and tasks.

2 Preliminaries

We ﬁrst introduce some point process terminology, most of which can be found in Chapters

8 and 9 from [34]. We work on Rd,d≥1, equipped with the Borel σ-ﬁeld Bdand Lebesgue

measure λd. Let Bd

bbe the bounded elements of Bd, and write Nfor the system of locally

ﬁnite subsets of Rd. For µ∈Nand B⊂Rdwe denote by µ(B) the cardinality of µ∩B.

By deﬁnition, we have µ(B)∈N0for each B∈ Bd

b. Let Nbe the smallest σ-ﬁeld on N

that renders the mappings µ7→ µ(B) measurable for each B∈ Bd.

A (simple) point process is a random element ηof (N,N), deﬁned on some ﬁxed

probability space (Ω,F,P) with associated expectation operator E. The process is called

stationary if the distribution of η+x:= {y+x:y∈η}does not depend on x∈Rd. In

this case γ:= Eη([0,1]d) is called the intensity or – in the terminology of physics – the

number density of η. By Campbell’s formula,

x∈η

f(x) = γZf(x) dx(2.1)

for each measurable function f:Rd→[0,∞). Here and in what follows, each unspeciﬁed

integral is over the full domain.

We consider a stationary point process ηwith positive and ﬁnite intensity γ. We

assume that ηis locally square integrable, i.e., Eη(B)2<∞for each B∈ Bd

b. The reduced

second factorial moment measure α!

2of ηis deﬁned by the formula

α!

2(·) := EX6=

x,y∈η

1x∈[0,1]d, y −x∈ ·.

Here, the superscript 6= indicates summation over pairs with diﬀerent components, and

1{·} stands for the indicator function. Notice that α!

2is a locally ﬁnite measure on Rd,

which is invariant under reﬂections at the origin and satisﬁes the equation

EX6=

x,y∈η

1{(x, y)∈ ·} =ZZ 1{(x, x +y)∈ ·}α!

2(dy) dx. (2.2)

Quite often the measure α!

2has a density (w.r.t. Lebesgue measure), which gives

α!

2(dy) = γ2g2(y) dy,

where g2:Rd→[0,∞) is the so-called pair correlation function of η. This function is

measurable and locally integrable, and it can be assumed to satisfy g2(x) = g2(−x) for

each x∈Rd. Given B∈ Bd, it follows from (2.2) that

Eη(B)2=γλd(B) + ZZ 1{x∈B, x +y∈B}dx α!

2(dy);

see [34, (4.25) and (8.8)]. If Bis bounded, then the variance of η(B) is

Var η(B) = γλd(B) + Zλd(B∩(B+y)) α!

2(dy)−γ2λd(B)2.

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AgenuinetestforhyperuniformityMichaelA.Klatt*,GunterLastandNorbertHenzeOctober25,2022AbstractWedevisetherstrigoroussignicancetestforhyperuniformitywithsensitivere-sults,evenforasinglesample.OurstartingpointisadetailedstudyoftheempiricalFouriertransformofastationarypointprocessonRd.Forlargesyste...

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