A joint work with G. Mastrilli, B. Blaszczyszyn, F. Lavancier
Hyperuniformity characterizes random spatial structures whose large-scale variance grows more slowly than that of Poisson processes. First introduced in statistical physics by Torquato and Stillinger [1], hyperuniform systems have attracted considerable attention due to their intermediate nature between perfect crystals, liquids, and glasses, and appears in a wide range of applications, from DNA organization and immune system dynamics to photoreceptor arrangements, urban planning, and cosmology.
Detecting and quantifying hyperuniformity is essential across these diverse fields. Yet, statistical testing for hyperuniformity has only recently begun to develop. In joint work [2], we address the problem of estimating the “strength” of hyperuniformity—formally, the exponent governing the decay of the spectral density near zero frequency—in a class of stationary point
processes in Euclidean space. The key mathematical idea is that the variance of linear statistics, defined using smooth, rapidly decaying test functions, grows in a way that explicitly reflects this exponent. Using a multivariate central limit theorem for a family of such statistics, constructed from orthogonal functions at multiple scales (e.g., wavelets), we derive an
asymptotically consistent estimator of the strength of hyperuniformity. This estimator can be computed from a single realization of the point process and comes with explicit confidence intervals. We validate this approach through simulations of various point process models and demonstrate its applicability on real data.
Bibliography
[1] Torquato, S. and Stillinger, F. H. (2003). Local density fluctuations, hyperuniformity, and order metrics. Physical Review E 68 041113.
[2] Mastrilli, G., B laszczyszyn, B. and Lavancier, F. (2024) Estimating the hyperuniformity exponent of point processes arXiv:2407.16797.