Hyperuniformity

Prof. Bartłomiej Błaszczyszyn wizytuje nasz instytut w ramach programu "profesorów wizytujących" IDUB. Hyperuniformity characterizes random spatial structures that exhibit slower growth in variance at large scales compared to Poisson structures emerging as a consequence of the complete independence assumption. Initially conceptualized in statistical physics by Torquato and Stillinger (2003), hyperuniform systems have garnered significant interest due to their unique position between perfect crystals, liquids, and glasses. These distinctive properties make them valuable for designing innovative materials, Monte Carlo numerical integration, and they have gained attention in various applied contexts, offering insights into phenomena ranging from DNA and the immune system to photoreceptors, urban systems, and cosmology. Detecting and quantifying hyperuniformity is crucial across these diverse domains. Despite this, statistical tests for hyperuniformity have only recently gained attention. In ongoing work with F. Lavancier and G. Mastrilli, we address the challenge of estimating the "strength" of hyperuniformity, related to the exponent of the spectrum at zero wavelength, in a class of stationary point processes in Euclidean. The mathematical core concept is that the variance of linear statistics, which are based on smooth and rapidly decreasing functions, grows in a way that explicitly involves this exponent. By leveraging the multivariate central limit theorem for many of these statistics, which involve orthogonal functions at different scales (wavelets), one obtains an asymptotically consistent estimator of the "strength" of hyperuniformity, based on a single realization of the point process, with explicit confidence intervals. We investigate this approach through numerical simulations using point process models and a real dataset.