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

The study of spatial point processes has evolved from simple Poisson models to complex determinantal and Gibbs processes, capturing phenomena like repulsion, clustering, and long-range order. The structure factor, originating from physics, quantifies spatial correlations in the frequency domain, serving as a key diagnostic for order versus randomness. Recent advances include the analysis of hyperuniformity, a state where density fluctuations diminish at large scales, relevant in materials science for designing novel materials with unique optical and mechanical properties. Despite its importance, the estimation of structure factors in finite samples remains challenging, especially under non-ideal conditions. Prior works, such as Torquato (2018) and Bardenet et al. (2020), provided theoretical insights and heuristic methods but lacked comprehensive statistical validation. This gap limited the reliability of hyperuniformity diagnostics in empirical studies, motivating the need for rigorous estimators and hypothesis tests.

Accurately estimating the structure factor from finite point samples is hindered by bias, variance, and boundary effects. Existing methods often rely on heuristic corrections or graphical inspection, lacking formal statistical guarantees. The challenge intensifies in high-dimensional or large-scale datasets, where computational efficiency and robustness are critical. Moreover, establishing a formal hypothesis test for hyperuniformity that is asymptotically valid remains an open problem. Addressing these issues is essential for advancing spatial analysis, enabling objective classification of point processes, and facilitating applications in material design, physics, and biology. The core difficulty lies in balancing estimator bias, variance, and computational cost while ensuring theoretical rigor and practical usability.

This work introduces a unified framework for structure factor estimation, combining classical scattering intensity, Bartlett’s estimator, Ogata quadrature, and Hankel transform methods, all enhanced with multiscale bias correction. It rigorously analyzes the asymptotic properties, establishing consistency and normality. The key innovation is the development of a formal, asymptotically valid hyperuniformity test based on these estimators, leveraging recent advances in debiasing techniques. The software implementation in Python encapsulates these algorithms, supporting automated, reproducible analysis. The approach bridges the gap between theoretical spectral analysis and practical hypothesis testing, providing a comprehensive toolkit for spatial structure diagnostics.

• �� Define the structure factor as the Fourier transform of the pair correlation measure, incorporating the scaled intersection volume. • Derive estimators based on empirical Fourier transforms of point configurations, including classical scattering intensity and multitaper variants. • Apply bias correction techniques, such as multiscale debiasing, to reduce finite-sample bias, especially near zero frequency. • Use numerical quadrature methods (Ogata, Baddour-Chouinard) to approximate Hankel transforms for isotropic cases. • Establish asymptotic properties (consistency, normality) through theoretical analysis, supported by simulations. • Develop a hypothesis testing procedure based on the asymptotic distribution of estimators, controlling type I error. • Implement algorithms in Python, supporting flexible data input and visualization, with modular design for extensibility.

Simulations involve generating point patterns from Ginibre, Poisson, Thomas, and thinned Ginibre processes, varying sample sizes from 10^3 to 10^5. Estimators are evaluated in terms of bias, variance, and mean squared error, focusing on low-frequency behavior. The hyperuniformity test’s power and size are assessed via repeated sampling, comparing against known process classes. Additional experiments analyze the robustness of estimators under boundary effects and anisotropy. Real data applications include material microstructure images and cosmic point distributions, validating the practical utility. Hyperparameter sensitivity, such as window size and taper selection, is systematically studied to optimize performance.

Estimates show that bias correction reduces low-frequency bias by over 50%, significantly improving the detection of hyperuniformity. The hyperuniformity test achieves >95% power at sample size 10^4, with controlled false positive rate. Structure factor estimates in Ginibre processes closely match theoretical S(k)=1−exp(−k^2/4), with errors below 0.01. Across different models, the estimators reliably distinguish hyperuniform (Ginibre) from non-hyperuniform (Poisson) processes. The software demonstrates high reproducibility and efficiency, supporting large datasets. Results confirm the decay rate of S(k) near zero as a key indicator, aligning with theoretical classifications into hyperuniform classes I and II.

The methods are directly applicable to analyzing microstructural arrangements in advanced materials, where controlling spatial order impacts optical and mechanical properties. In astrophysics, they enable quantification of galaxy or star distributions, aiding in understanding large-scale structure formation. The software supports spatial pattern recognition in biological tissues, ecological surveys, and urban planning. Long-term, integrating these estimators with machine learning could automate structure classification, facilitating real-time analysis of complex spatial data in scientific and industrial contexts.

Assumptions of stationarity and isotropy limit applicability to real-world data with anisotropic or non-stationary features. Computational complexity increases with dimension and dataset size, requiring further optimization. Finite sample sizes restrict estimator accuracy and statistical power, especially for subtle hyperuniformity signals. The current framework does not address non-ergodic or dynamic point processes, which are common in practice. Future work should focus on relaxing assumptions, improving scalability, and extending to non-stationary models.