Spectral leakage and masking effects in the measurement of hyperuniformity

Disordered hyperuniformity is a recently discovered novel state of matter, characterized by a complete suppression of normalized infinite-wavelength density fluctuations as in perfect crystals and lack of conventional long-range order nor broken symmetry as in glasses. The detection of hyperuniformity relies critically on accurate characterization of the small-wavenumber behavior of the static structure factor of the system. In practice, however, measurements are performed on finite subsystems or through incomplete observations that effectively mask portions of the underlying configuration. Inspired by a recent numerical study [Liu et al., J. Chem. Phys. 164, 094102 (2026)], we develop a unified theoretical framework that quantifies how finite windows and spatially correlated binary masks modify the observed structure factor. We show that the measured structure factor Sobs(k) is the convolution of the intrinsic structure factor with the spectral density of the observation function, whether it is a compact window or an extended random mask. For generic hyperuniform systems with small-k scaling S(k) ∼ kα, finite observation window induces a universal quadratic leakage term at sufficiently small wavenumbers (i.e., k ≲ 1/L), leading to an apparent k2 scaling independent of the true exponent. The true hyperuniform exponent α can only be measured in the intermediate regime 1/L ≪ k ≪ qc. In stealthy hyperuniform systems, where the intrinsic structure factor possesses a spectral gap, all observed small-k power arises entirely from this convolution mechanism. For spatially correlated masks, we derive the corresponding convolution relation in terms of the mask spectral density and identify conditions under which hyperuniform signatures are suppressed, preserved, or distorted. Our results establish quantitative criteria for reliably extracting intrinsic scaling exponents and distinguishing genuine hyperuniform order from measurement-induced artifacts.