Our systematic study of local number fluctuations in both hyperuniform and nonhyperuniform systems has revealed some intriguing behaviors and provides useful tools for further studies:

S. Torquato, J. Kim, M. A. Klatt. Local Number Fluctuations in Hyperuniform and Nonhyperuniform Systems: Higher-Order Moments and Distribution Functions. *Phys. Rev. X* **11**:021028-1–27 (2021).

**Popular Summary by Physical Review X:**

“The emerging field of ‘hyperuniformity’ provides mathematical tools for classifying large ensembles of particles based on the degree to which density fluctuations are suppressed according to their first two moments (the expected value and the variance). Originally formulated to describe how atoms are positioned in crystals, quasicrystals, and amorphous solids, the framework is useful in many studies, from the distribution of prime numbers to photoreceptor cells in avian retina. Here, we stress the importance of characterizing the higher-order moments of any system, hyperuniform or not.

To more completely characterize density fluctuations, we carry out an extensive theoretical and computational study of higher-order moments and the corresponding probability distribution function of a large class of models across the first three space dimensions. These models describe both hyperuniform and nonhyperuniform systems, that is, those in which density fluctuations are greatly suppressed and those in which they are not. Remarkably, we discover that diverse systems—encompassing the majority of our models—can be described by ‘universal’ distribution functions.

Our work elucidates the fundamental importance of higher-order structural information to fully characterize density fluctuations in many-body systems across length scales and dimensions, and thus it has broad implications for condensed-matter physics, engineering, mathematics, and biology.”

From https://link.aps.org/doi/10.1103/PhysRevX.11.021028

**A few more details:**

We derived rigorous bounds on higher-moments of the number distributions, beyond the number variance, and we developed a robust Gaussian “distance” metric for number distributions. We inferred the presence of three- and higher-body correlations in some of our models by the behaviors of the higher-order moments. Moreover, we showed for models that decorrelate or correlate, as the space dimension increases, that fluctuations become increasingly Gaussian or non-Gaussian, respectively. Finally, we found a universal behavior at intermediate sizes of the observation windows in that the number distributions are well approximated by gamma distributions for all of our distinctly different models that obey a CLT. For our models, we thus found that the convergence to a CLT is slower for standard nonhyperuniform models and slowest for the ‘antihyperuniform’ model studied here.

**Supplementary dataset:**

We have published all results of our analysis on Zenodo:

https://doi.org/10.5281/zenodo.4635977