  We numerically analyse the density field of three-dimensional randomly jammed
packings of monodisperse soft frictionles spherical particles, paying special
attention to fluctuations occurring at large lengthscales. We study in detail
the two-point static structure factor at low wavevectors in Fourier space. We
also analyse the nature of the density field in real space by studying the
large-distance behavior of the two-point pair correlation function, of density
fluctuations in subsystems of increasing sizes, and of the direct correlation
function. We show that such real space analysis can be greatly improved by
introducing a coarse-grained density field to disentangle genuine large-scale
correlations from purely local effects. Our results confirm that both Fourier
and real space signatures of vanishing density fluctuations at large scale are
absent, indicating that randomly jammed packings are not hyperuniform. In
addition, we establish that the pair correlation function displays a
surprisingly complex structure at large distances, which is however not
compatible with the long-range negative correlation of hyperuniform systems but
fully compatible with an analytic form for the structure factor. This implies
that the direct correlation function is short-ranged, as we also demonstrate
directly. Our results reveal that density fluctuations in jammed packings do
not follow the behavior expected for random hyperuniform materials, but display
instead a more complex behavior.
