  As a function of packing fraction at zero temperature and applied stress, an
amorphous packing of spheres exhibits a jamming transition where the system is
sensitive to boundary conditions even in the thermodynamic limit. Upon further
compression, the system should become insensitive to boundary conditions
provided it is sufficiently large. Here we explore the linear response to a
large class of boundary perturbations in 2 and 3 dimensions. We consider each
finite packing with periodic-boundary conditions as the basis of an infinite
square or cubic lattice and study properties of vibrational modes at arbitrary
wave vector. We find that the stability of such modes be understood in terms of
a competition between plane waves and the anomalous vibrational modes
associated with the jamming transition; infinitesimal boundary perturbations
become irrelevant for systems that are larger than a length scale that
characterizes the transverse excitations. This previously identified length
diverges at the jamming transition.
