  We study theoretically and numerically how hard frictionless particles in
random packings can rearrange. We demonstrate the existence of two distinct
unstable non-linear modes of rearrangement, both associated with the opening
and the closing of contacts. Mode one, whose density is characterized by some
exponent {\theta}', corresponds to motions of particles extending throughout
the entire system. Mode two, whose density is characterized by an exponent
{\theta} != {\theta}', corresponds to the local buckling of a few particles.
Mode one is shown to yield at a much higher rate than mode two when a stress is
applied. We show that the distribution of contact forces follows P(f)
f^{min({\theta}',{\theta})}, and that imposing that the packing cannot be
densified further leads to the bounds {\gamma} >= 1/(2+{\theta}') and {\gamma}
>= (1-{\theta})/2, where {\gamma} characterizes the singularity of the pair
distribution function g(r) at contact. These results extend the theoretical
analysis of [M. Wyart, Phys. Rev. Lett 109, 125502 (2012)] where the existence
of mode two was not considered. We perform numerics that support that these
bounds are saturated with {\gamma} \approx 0.38, {\theta} \approx 0.17 and
{\theta}' \approx 0.44. We measure systematically the stability of all such
modes in packings, and confirm their marginal stability. The principle of
marginal stability thus allows to make clearcut predictions on the ensemble of
configurations visited in these out-of-equilibrium systems, and on the contact
forces and pair distribution functions. It also reveals the excitations that
need to be included in a description of plasticity or flow near jamming, and
suggests a new path to study two-level systems and soft spots in simple
amorphous solids of repulsive particles.
