Soft modes and the onset of jamming

Glasses have a large excess of low-frequency vibrational modes in comparison with crystalline solids. We show that such a feature is a necessary consequence of the geometry generic to a {\it marginally} connected solid. In particular, we analyze the density of states of a recently simulated system comprised of weakly compressed spheres at zero temperature. We account for the observed a) constancy of the density of modes with frequency, b) appearance of a low-frequency cutoff $\omega^*$, and c) power-law increase of $\omega^*$ with compression. We predict a length scale $l^*$ below which the boundary conditions strongly affect the system. $l^*$ diverges at the jamming transition when the system becomes isostatic.