Spectrum of structure for jammed and unjammed soft disks

We investigate the short-, medium-, and long-range structure of soft-disk configurations for a wide range of area fractions and simulation protocols by converting the real-space spectrum of volume fraction fluctuations for windows of width L to the distance h (L ) from the window boundary over which fluctuations occur. Rapidly quenched unjammed configurations exhibit size-dependent super-Poissonian long-range features that surprisingly approach the totally random limit even close to jamming. Above and just below jamming, the spectra exhibit a plateau h (L ) =h_e for L larger than particle size and smaller than a cutoff L_c beyond which there are long-range fluctuations. The value of h_e is independent of protocol and characterizes the putative hyperuniform limit. This behavior is compared with that for Einstein solids, with and without hyperuniformity-destroying defects. We find that key structural features of the particle configurations are more evident, as well as easier and more intuitive to quantify, using the real-space spectrum of hyperuniformity lengths rather than the spectral density.