Papers › Estimating the hyperuniformity exponent of point processes
Estimating the hyperuniformity exponent of point processes
Gabriel Mastrilli, Bartłomiej Błaszczyszyn, Frédéric Lavancier
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We address the challenge of estimating the hyperuniformity exponent α of a spatial point process, given only one realization of it. Assuming that the structure factor S of the point process follows a vanishing power law at the origin (the typical case of a hyperuniform point process), this exponent is defined as the slope near the origin of logS. Our estimator is built upon the (expanding window) asymptotic variance of some wavelet transforms of the point process. By combining several scales and several wavelets, we develop a multi-scale, multi-taper estimator α. We analyze its asymptotic behavior, proving its consistency under various settings, and enabling the construction of asymptotic confidence intervals for α when α< d and under Brillinger mixing. This construction is derived from a multivariate central limit theorem where the normalisations are non-standard and vary among the components. We also present a non-asymptotic deviation inequality providing insights into the influence of tapers on the bias-variance trade-off of α. Finally, we investigate the performance of α through simulations, and we apply our method to the analysis of hyperuniformity in a real dataset of marine algae.
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