  We consider a system of hard spheres close to jamming, where translation
invariance is broken by pinning a randomly chosen set of particles. Using two
different protocols, we generate two kinds of packings at the jamming point,
isostatic and hyperstatic packings. In the case where the packings are
isostatic, jamming transition is not affected by random pinning: the jamming
density is only slightly reduced, a generalized isostaticity condition holds,
the system is marginally rigid over the entire glass phase, and the typical
structural signatures of jamming are unchanged. Besides, random pinning does
not have effect on the vibrational modes of the amorphous system, at least on
the time and length scale that we are able to probe. For packings that are
hyperstatic at jamming, the microscopic properties of the system are strongly
affected by the random pinning: the frozen degrees of freedom introduce an
excess of constraints that drives the system far from the mechanical
marginality conditions and stabilize the low frequency modes of the system,
shifting them to higher frequencies. The distance from the isostatic point
seems to be the only relevant parameter in the system in both cases. These two
cases are in contrast with the behavior of the plane waves of a crystal: the
whole spectrum in this case is shifted to higher frequencies as soon as some
particles are pinned.
