Quantifying when hyperuniformity of a many-particle system leads to uniformity across length scales

Hyperuniform systems are distinguished by an unusually strong suppression of large-scale density fluctuations and, consequently, display a high degree of uniformity at the largest length scales. In some cases, however, enhanced uniformity is expected to be present even at intermediate and possibly small length scales. There exist three different classes of hyperuniform systems, where class I and class III are the strongest and weakest forms, respectively. We utilize the local number variance $\sigma_N^2(R)$ associated with a window of radius $R$ as a diagnostic to quantify the approach to the asymptotic large-$R$ hyperuniform scaling of a variety of class I, II, and III systems. We find, for all class I systems we analyzed, including crystals, quasicrystals, disordered stealthy hyperuniform systems, and the one-component plasma, a faster approach to the asymptotic scaling of $\sigma_N^2(R)$, governed by corrections with integer powers of $1/R$. Thus, we conclude this represents the highest degree of effective uniformity from small to large length scales. Class II systems, such as Fermi-sphere point processes, are characterized by logarithmic $1/\ln(R)$ corrections and, consequently, a lower degree of local uniformity. Class III systems, such as perturbed lattice patterns, present an asymptotic scaling of $1/R^{\alpha}$, $0 < \alpha < 1$, implying, curiously, an intermediate degree of local uniformity. In addition, our study provides insight into when experimental and numerical finite systems are representative of large-scale behavior. Our findings may thereby facilitate the design of hyperuniform systems with enhanced physical properties arising from local uniformity.