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

Understanding structural organization in disordered systems is central to soft and glassy matter physics. Hyperuniform materials form a unique class of disordered structures that, despite lacking periodicity, strongly suppress large-scale density fluctuations. In some cases, enhanced uniformity also extends to intermediate and small scales. Using the local number variance σ²ₙ(R) within a window of radius R, we quantify the approach to asymptotic hyperuniform scaling in representative class I, II, and III systems. For all class I systems examined—including crystals, quasicrystals, disordered stealthy hyperuniform structures, and the one-component plasma—we find a rapid approach to scaling governed by integer-power 1/R corrections, indicating maximal uniformity. Class II (Fermi-sphere point processes) exhibit logarithmic 1/ln(R) corrections, while class III (perturbed lattices) show 1/R^α behavior (0 < α < 1), signifying intermediate uniformity. These results clarify when finite systems capture large-scale behavior and guide the design of hyperuniform materials with enhanced physical properties.