Structural properties of hyperuniform Voronoi networks

Disordered hyperuniform many-particle systems are recently discovered exotic states of matter, characterized by the complete suppression of normalized infinite-wavelength density fluctuations, as in perfect crystals, while lacking conventional long-range order, as in liquids and glasses. In this work, we begin a program to quantify the structural properties of nonhyperuniform and hyperuniform networks. In particular, large two-dimensional (2D) Voronoi networks (graphs) containing approximately 10,000 nodes are created from a variety of different point configurations, including the antihyperuniform hyperplane intersection process (HIP), nonhyperuniform Poisson process, nonhyperuniform random sequential addition (RSA) saturated packing, and both non-stealthy and stealthy hyperuniform point processes. We carry out an extensive study of the Voronoi-cell area distribution of each of the networks by determining multiple metrics that characterize the distribution, including their average areas and corresponding variances as well as higher-order cumulants (i.e., skewness γ_{1} and excess kurtosis γ_{2}). We show that the HIP distribution is far from Gaussian, as evidenced by a high skewness (γ_{1}=3.16) and large positive excess kurtosis (γ_{2}=16.2). The Poisson (with γ_{1}=1.07 and γ_{2}=1.79) and non-stealthy hyperuniform (with γ_{1}=0.257 and γ_{2}=0.0217) distributions are Gaussian-like distributions, since they exhibit a small but positive skewness and excess kurtosis. The RSA (with γ_{1}=0.450 and γ_{2}=-0.0384) and the highest stealthy hyperuniform distributions (with γ_{1}=0.0272 and γ_{2}=-0.0626) are also non-Gaussian because of their low skewness and negative excess kurtosis, which is diametrically opposite of the non-Gaussian behavior of the HIP. The fact that the cell-area distributions of large, finite-sized RSA and stealthy hyperuniform networks (e.g., with N≈10,000 nodes) are narrower, have larger peaks, and smaller tails than a Gaussian distribution implies that in the thermodynamic limit the distributions should exhibit compact support, consistent with previous theoretical considerations. Moreover, we compute the Voronoi-area correlation functions C_{00}(r) for the networks, which describe the correlations between the area of two Voronoi cells separated by a given distance r [M. A. Klatt and S. Torquato, Phys. Rev. E 90, 052120 (2014)1539-375510.1103/PhysRevE.90.052120]. We show that the correlation functions C_{00}(r) qualitatively distinguish the antihyperuniform, nonhyperuniform, and hyperuniform Voronoi networks considered here. Specifically, the antihyperuniform HIP networks possess a slowly decaying C_{00}(r) with large positive values, indicating large fluctuations of Voronoi cell areas across scales. While the nonhyperuniform Poisson and RSA network possess positive and fast decaying C_{00}(r), we find strong anticorrelations in C_{00}(r) (i.e., negative values) for the hyperuniform networks. The latter indicates that the large-scale area fluctuations are suppressed by accompanying large Voronoi cells with small cells (and vice versa) in the systems in order to achieve hyperuniformity. In summary, we have shown that cell-area distributions and pair correlation functions of Voronoi networks enable one to distinguish quantitatively antihyperuniform, standard nonhyperuniform, and hyperuniform networks from one another.