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On estimating the structure factor of a point process, with applications to hyperuniformity


Hyperuniformity is the study of stationary point processes with a sub-Poisson variance in a large window. In other words, counting the points of a hyperuniform point process that fall in a given large region yields a small-variance Monte Carlo estimation of the volume. Hyperuniform point processes have received a lot of attention in statistical physics, both for the investigation of natural organized structures and the synthesis of materials. Unfortunately, rigorously proving that a point process is hyperuniform is usually difficult. A common practice in statistical physics and chemistry is to use a few samples to estimate a spectral measure called the structure factor. Its decay around zero provides a diagnostic of hyperuniformity. Different applied fields use however different estimators, and important algorithmic choices proceed from each field’s lore. This paper provides a systematic survey and derivation of known or otherwise natural estimators of the structure factor. We also leverage the consistency of these estimators to contribute the first asymptotically valid statistical test of hyperuniformity. We benchmark all estimators and hyperuniformity diagnostics on a set of examples. In an effort to make investigations of the structure factor and hyperuniformity systematic and reproducible, we further provide the Python toolbox structure_factor, containing all the estimators and tools that we discuss.

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Code availability

The code is published as an open-source Python toolbox under the project name structure-factor. he package is licensed under the MIT license and is available on Github and PyPI



  2. E.g., Hypothesis (H4) of Biscio and Waagepetersen (2019), when the linear statistic is the number of points, contradicts hyperuniformity.

  3. We note that during the reviewing process of our paper, a second version of the preprint (Rajala et al. 2020a) has been arXived (Rajala et al. 2020b). The changes in the new version do not seem to impact our work.

  4. The literature is inconsistent as to whether the structure factor is the measure \({\mathcal {S}}\) or its density S. We choose the density, which is also sometimes known as the scaled spectral density function.




  8. At and on PyPI.

  9. A violin plot gathers a box plot and a kernel density estimator of the assumed underlying density. The former shows the median (white point), the interquartile range (thick black bar in the center), and the rest of the distribution except for points determined as outliers (thin black line in the center). We also add the mean (red point).


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We thank Jean-François Coeurjolly, Michael Andreas Klatt, Günther Last, David Dereudre, and Simon Coste for insightful discussions along this project. The motivation for our multiscale test was sparked at the Karlsruhe workshop on New trends in point process theory in March 2022, by the lively discussions with other participants about a preliminary draft of our paper. We know from personal communication that, by then, Günther Last, Andreas Klatt, and Norbert Henze were independently working on their own test, whose preprint Anderson et al. (2009) came out as we were answering the referees on our own manuscript. Finally, we thank a referee for suggesting thinning a hyperuniform point process to assess the detection performance of our test. It has been independently brought to our attention that this procedure can also be found in Baddeley et al. (2015).


This work is supported by ERC-2019-STG-851866 and ANR-20-CHIA-0002.

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A Proof of Proposition 3


Let \(M \in L^p\) with \(p>0\). We first prove that \(Z_m\rightarrow Z\) in \(L^p\). As we have \(Z_m \rightarrow Z\) a.s., it is enough to show that \(Z_m\) is uniformly bounded in \(L^p\). For a realization \(M'\) of M we have,

$$\begin{aligned} |Z_m |\le \sum _{j=1}^{m \wedge M'} \frac{|Y_{j} - Y_{j-1} |}{{\mathbb {P}} (M \ge j)} \le \frac{M'}{{\mathbb {P}}(M \ge M')}. \end{aligned}$$

By assumption \(M \in L^p\) so \(Z_m\) is uniformly bounded in \(L^p\). This proves the first part of the proposition.

Before proving the additional two points, note that, since S is continuous,

$$\begin{aligned} {\mathbb {E}} [\widehat{S}_m(\textbf{k}_m^{\textrm{min}})] \xrightarrow [m\rightarrow \infty ]{} S(\textbf{0}). \end{aligned}$$

Now, let us prove the first point of the proposition. Assume that \(M \in L^1\) and \({\mathcal {X}}\) is hyperuniform, so that \(S(\textbf{0} )= 0\). Since \(\hat{S}_m\) is nonnegative, Equation (57) yields

$$\begin{aligned} \widehat{S}_m(\textbf{k}_m^{\textrm{min}}) \xrightarrow [m\rightarrow \infty ]{ L^1} 0. \end{aligned}$$

Moreover, letting \(f:x\mapsto 1\wedge x\), \(\vert f(x)\vert \le x\) on \({\mathbb {R}}^+\), so that

$$\begin{aligned} {\mathbb {E}} [\vert f(\widehat{S}_m(\textbf{k}_m^{\textrm{min}}))\vert ] \le {\mathbb {E}} [ \widehat{S}_m(\textbf{k}_m^{\textrm{min}}) ]\rightarrow 0, \end{aligned}$$


$$\begin{aligned} Y_m = f(\widehat{S}_m(\textbf{k}_m^{\textrm{min}})) \xrightarrow [m\rightarrow \infty ]{ L^1} 0. \end{aligned}$$

Since \({\mathbb {E}} [Y_m]= {\mathbb {E}}[Z_m]\) and \(Z_m\) converges in \(L^1\) to Z, by unicity of the limit, we have \({\mathbb {E}} [Z]=0\).

It remains to show the last point of the proposition. Assume again that \(M \in L^1\), but that \({\mathcal {X}}\) is not hyperuniform, so that \(S( \textbf{0}) >0\). Reasoning by contradiction, assume that \({\mathbb {E}} [Z]=0\). As \({\mathbb {E}} [Y_m]= {\mathbb {E}}[Z_m]\) and \(Z_m\) converges in \(L^1\) to Z, we get

$$\begin{aligned} {\mathbb {E}} [\widehat{S}_m(k_m^{\textrm{min}}) \mathbbm {1}_{\{ \widehat{S}_m(k_m^{\textrm{min}})<1 \}}] \xrightarrow [m \rightarrow \infty ]{} 0, \end{aligned}$$


$$\begin{aligned} {\mathbb {E}} [ \mathbbm {1}_{\{ \widehat{S}_m (k_m^{\textrm{min}}) \ge 1 \}}] \xrightarrow [m \rightarrow \infty ]{} 0. \end{aligned}$$


$$\begin{aligned} {\mathbb {E}} [\widehat{S}_m(k_m^{\textrm{min}})] \xrightarrow [m \rightarrow \infty ]{} S(\textbf{0} ) >0. \end{aligned}$$

Using Eq. (59), we get

$$\begin{aligned} {\mathbb {E}} [\widehat{S}_m(k_m^{\textrm{min}}) \mathbbm {1}_{\{ \widehat{S}_m(k_m^{\textrm{min}}) \ge 1 \}}] \xrightarrow [m \rightarrow \infty ]{} S(\textbf{0}) >0. \end{aligned}$$

Finally, Cauchy–Schwarz, together with Condition (54) and Eq. (60) yield

$$\begin{aligned}&{\mathbb {E}} [\widehat{S}_m(k_m^{\textrm{min}}) \mathbbm {1}_{\{ \widehat{S}_m(k_m^{\textrm{min}}) \ge 1 \}}] \le \\&\quad {\mathbb {E}} ^{1/2}[\widehat{S}^2_m(k_m^{\textrm{min}})] \times {\mathbb {E}}^{1/2} [\mathbbm {1}_{\{ \widehat{S}_m (k_m^{\textrm{min}}) \ge 1 \}}] \rightarrow 0, \end{aligned}$$

which contradicts Eq. (61) and ends the proof. \(\square \)

B Validity of Assumption (54)

In what follows, we show that Assumption (54) is satisfied for a homogeneous Poisson point process \({\mathcal {X}}\) of intensity \(\rho \), \(\widehat{S}= \widehat{S}_{\textrm{SI}}\), and \(W_m\) are increasing rectangular windows.

Let \(N_m = |{{\mathcal {X}} \cap W_m}|\). Then

$$\begin{aligned} \rho ^2&|{W_m}|^2 {\mathbb {E}} [\widehat{S}^2_m(\textbf{k}) ]\\&= {\mathbb {E}} \left( \left|\sum _{\textbf{x} \in {\mathcal {X}} \cap W_m } e^{- \textrm{i}\langle \textbf{k}, \textbf{x} \rangle } \right|^2 \right) ^2 \\&= {\mathbb {E}} \Big [ N_m^2 + \big (\sum _{\textbf{x}, \textbf{y} \in {\mathcal {X}} \cap W_m }^{\textbf{x} \ne \textbf{y}} e^{- \textrm{i}\langle \textbf{k}, \textbf{x} - \textbf{y} \rangle } \big ) ^2 \\&\quad + 2 N_m \sum _{\textbf{x}, \textbf{y} \in {\mathcal {X}} \cap W_m }^{\textbf{x} \ne \textbf{y}} e^{- \textrm{i}\langle \textbf{k}, \textbf{x} - \textbf{y} \rangle } \Big ] \\&= {\mathbb {E}} \Big [ 2 N_m^2 - N_m \\&\quad + \sum _{\textbf{x}, \textbf{y} \in {\mathcal {X}} \cap W_m }^{\textbf{x} \ne \textbf{y}} 2 N_me^{- \textrm{i}\langle \textbf{k}, \textbf{x} - \textbf{y} \rangle } + e^{- \textrm{i}2 \langle \textbf{k}, \textbf{x} - \textbf{y} \rangle }\\&\quad + \sum _{\textbf{x}, \textbf{y}, \textbf{z} \in {\mathcal {X}} \cap W_m }^{\textbf{x} \ne \textbf{y} \ne \textbf{z}} 2 e^{- \textrm{i}\langle \textbf{k}, \textbf{x} - \textbf{y} \rangle } + e^{- \textrm{i}\langle \textbf{k}, 2 \textbf{x} - \textbf{y} - \textbf{z} \rangle } + e^{ \textrm{i}\langle \textbf{k}, 2 \textbf{x} - \textbf{y} - \textbf{z} \rangle } \\&\quad + \sum _{\textbf{x}, \textbf{y}, \textbf{z}, \textbf{t} \in {\mathcal {X}} \cap W_m }^{\textbf{x} \ne \textbf{y} \ne \textbf{z} \ne \textbf{t}} e^{- \textrm{i}\langle \textbf{k}, \textbf{x} - \textbf{y} + \textbf{z} - \textbf{t}\rangle }\Big ]\\&= {\mathbb {E}} \Big [ 2 N_m^2 - N_m +\sum _{\textbf{x}, \textbf{y} \in {\mathcal {X}} \cap W_m }^{\textbf{x} \ne \textbf{y}} 4 e^{- \textrm{i}\langle \textbf{k}, \textbf{x} - \textbf{y} \rangle } + e^{- \textrm{i}2 \langle \textbf{k}, \textbf{x} - \textbf{y} \rangle } \\&\quad + \sum _{\textbf{x}, \textbf{y}, \textbf{z} \in {\mathcal {X}} \cap W_m }^{\textbf{x} \ne \textbf{y} \ne \textbf{z}} 4 e^{- \textrm{i}\langle \textbf{k}, \textbf{x} - \textbf{y} \rangle } + e^{- \textrm{i}\langle \textbf{k}, 2 \textbf{x} - \textbf{y} - \textbf{z} \rangle } + e^{ \textrm{i}\langle \textbf{k}, 2 \textbf{x} - \textbf{y} - \textbf{z} \rangle } \\&\quad + \sum _{\textbf{x}, \textbf{y}, \textbf{z}, \textbf{t} \in {\mathcal {X}} \cap W_m }^{\textbf{x} \ne \textbf{y} \ne \textbf{z} \ne \textbf{t}} e^{- \textrm{i}\langle \textbf{k}, \textbf{x} - \textbf{y} + \textbf{z} - \textbf{t}\rangle } \Big ] \\&= {\mathbb {E}} [2 N_m^2 - N_m] \\&\quad + \int _{W_m \times W_m} \Big (4 e^{- \textrm{i}\langle \textbf{k}, \textbf{x} - \textbf{y} \rangle } + e^{- \textrm{i}2 \langle \textbf{k}, \textbf{x} - \textbf{y} \rangle } \Big ) \rho ^2 \textrm{d}{} \textbf{x} \textrm{d}{} \textbf{y} \\&\quad + \int _{W_m \times W_m \times W_m} \Big (4 e^{- \textrm{i}\langle \textbf{k}, \textbf{x} - \textbf{y} \rangle } + e^{- \textrm{i}\langle \textbf{k}, 2 \textbf{x} - \textbf{y} - \textbf{z} \rangle } \\&\quad + e^{ \textrm{i}\langle \textbf{k}, 2 \textbf{x} - \textbf{y} - \textbf{z} \rangle } \Big ) \rho ^3 \textrm{d}{} \textbf{x} \textrm{d}{} \textbf{y} \textrm{d}{} \textbf{z}\\&\quad + \int _{W_m \times W_m \times W_m \times W_m} e^{- \textrm{i}\langle \textbf{k}, \textbf{x} - \textbf{y} + \textbf{z} - \textbf{t}\rangle } \rho ^4 \textrm{d}{} \textbf{x} \textrm{d}{} \textbf{y} \textrm{d}{} \textbf{z} \textrm{d}{} \textbf{t}. \end{aligned}$$

The last line was obtained using the definition of the n-th product density \(\rho ^{(n)}\) and that for any \(n \ge 1\), \(\rho ^{(n)}\) of \({\mathcal {X}}\) simplifies to \(\rho ^n\). It is a well-known property of homogeneous Poisson point processes (Chiu et al. 2013, Section 2.3.3). Now, using the parity of \(\mathbbm {1}_{W_m}\) and that \(N_m\) is a Poisson r.v., we get

$$\begin{aligned}&{\mathbb {E}} [\widehat{S}^2_m(\textbf{k}_m^{\textrm{min}}) ]\nonumber \\&\quad = \frac{1}{(\rho |{W_m}|)^2 } \Big [ \rho |{W_m}| + 2 (\rho |{W_m}|)^2 \nonumber \\&\qquad + \rho ^4 {\mathcal {F}}^4 (\mathbbm {1}_{W_m}) (\textbf{k}_m^{\textrm{min}}) \nonumber \\&\qquad + \rho ^2 \Big (4 {\mathcal {F}}^2(\mathbbm {1}_{W_m}) (\textbf{k}_m^{\textrm{min}}) + {\mathcal {F}}^2(\mathbbm {1}_{W_m}) (2 \textbf{k}_m^{\textrm{min}})\Big ) \nonumber \\&\qquad + \rho ^3 \Big ( 4 |{W_m}| {\mathcal {F}}^2 (\mathbbm {1}_{W_m})(\textbf{k}_m^{\textrm{min}}) \nonumber \\&\qquad + 2 {\mathcal {F}}(\mathbbm {1}_{W_m}) (2 \textbf{k}_m^{\textrm{min}}) {\mathcal {F}}^2(\mathbbm {1}_{W_m}) (\textbf{k}_m^{\textrm{min}}) \Big ) \Big ]. \end{aligned}$$

Upon noting that \(\textbf{k}_m^{\textrm{min}} = (\frac{2 \pi }{L_1}, \cdots , \frac{2 \pi }{L_d})\) and

$$\begin{aligned} {\mathcal {F}}(\mathbbm {1}_{W_m})(\textbf{k})= \prod _{j=1}^d \frac{\sin (k_j L_j/2)}{k_j/2}. \end{aligned}$$

Equation (62) simplifies to

$$\begin{aligned} {\mathbb {E}} [\widehat{S}^2_m(\textbf{k}_m^{\textrm{min}}) ] = \frac{1}{\rho |{W_m}|} + 2. \end{aligned}$$

Thus Assumption (54) holds.

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Hawat, D., Gautier, G., Bardenet, R. et al. On estimating the structure factor of a point process, with applications to hyperuniformity. Stat Comput 33, 61 (2023).

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