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# “Local Number Fluctuations in Hyperuniform and Nonhyperuniform Systems: Higher-Order Moments and Distribution Functions” is Published in Physical Review X

The local number variance $$\sigma^2(R)$$ associated with a spherical sampling window of radius $$R$$ enables a classification of many-particle systems in $$d$$-dimensional Euclidean space $$\mathbb{R}^d$$ according to the degree to which large-scale density fluctuations are suppressed, resulting in a demarcation between hyperuniform and nonhyperuniform phyla. To more completely characterize density fluctuations, we carry out an extensive study of higher-order moments or cumulants, including the skewness $$\gamma_1(R)$$, excess kurtosis $$\gamma_2(R)$$, and the corresponding probability distribution function $$P[N(R)]$$ of a large family of models across the first three space dimensions, including both hyperuniform and nonhyperuniform systems with varying degrees of short- and long-range order. To carry out this comprehensive program, we derive new theoretical results that apply to general point processes, and we conduct high-precision numerical studies. Specifically, we derive explicit closed-form integral expressions for $$\gamma_1(R)$$ and $$\gamma_2(R)$$ that encode structural information up to three-body and four-body correlation functions, respectively. We also derive rigorous bounds on $$\gamma_1(R)$$, $$\gamma_2(R)$$, and $$P[N(R)]$$ for general point processes and corresponding exact results for general packings of identical spheres. High-quality simulation data for $$\gamma_1(R)$$, $$\gamma_2(R)$$, and $$P[N(R)]$$ are generated for each model. We also ascertain the proximity of $$P[N(R)]$$ to the normal distribution via a novel Gaussian “distance” metric $$l_2(R)$$. Among all models, the convergence to a central limit theorem (CLT) is generally fastest for the disordered hyperuniform processes in two or higher dimensions such that $$\gamma_1(R)\sim l_2(R)\sim R^{-(d+1)/2}$$ for large $$R$$. The convergence to a CLT is slower for standard nonhyperuniform models and slowest for the “antihyperuniform” model studied here. We prove that one-dimensional hyperuniform systems of class I or any $$d$$-dimensional lattice cannot obey a CLT. Remarkably, we discover a type of universality in that, for all of our models that obey a CLT, the gamma distribution provides a good approximation to $$P[N(R)]$$ across all dimensions for intermediate to large values of $$R$$, enabling us to estimate the large-$$R$$ scalings of $$\gamma_1(R)$$, $$\gamma_2(R)$$, and $$l_2(R)$$. For any $$d$$-dimensional model that “decorrelates” or “correlates” with $$d$$, we elucidate why $$P[N(R)]$$ increasingly moves toward or away from Gaussian-like behavior, respectively. Our work sheds light on the fundamental importance of higher-order structural information to fully characterize density fluctuations in many-body systems across length scales and dimensions, and thus has broad implications for condensed matter physics, engineering, mathematics, and biology.