  We numerically study the structure of the interactions occurring in
three-dimensional systems of hard spheres at jamming, focusing on the
large-scale behavior. Given the fundamental role they play in the configuration
of jammed packings, we analyze the propagation through the system of the weak
forces and of the variation of the coordination number with respect to the
isostaticity condition $\Delta Z$. We show that these correlations can be
successfully probed by introducing a correlation function weighted on the
density-density fluctuations. The results of this analysis can be further
improved by introducing a representation of the system based on the contact
points between particles. In particular, we find evidence that the weak forces
and the $\Delta Z$ fluctuations support the hypothesis of randomly jammed
packings of spherical particles being hyperuniform by exhibiting an anomalous
long-range decay. Moreover, we find that the large-scale structure of the
density-density correlation exhibits a complex behavior due to the
superimposition of two exponentially damped oscillating signals propagating
with linearly depending frequencies.
