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

Abstract

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.

Appendices

A Proof of Proposition 3

Proof

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,

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

(57)

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

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

and

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

(58)

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

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

(59)

and

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

(60)

Meanwhile,

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

Using Eq. (59), we get

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

(61)

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

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

(62)

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.