  We study the pair correlation function $g(r)$ for zero-temperature,
disordered, soft-sphere packings just above the onset of jamming. We find
distinct signatures of the transition in both the first and split second peaks
of this function. As the transition is approached from the jammed side (at
higher packing fraction) the first peak diverges and narrows on the small-$r$
side to a delta-function. On the high-$r$ side of this peak, $g(r)$ decays as a
power-law. In the split second peak, the two subpeaks are both singular at the
transition, with power-law behavior on their low-$r$ sides and step-function
drop-offs on their high-$r$ sides. These singularities at the transition are
reminiscent of empirical criteria that have previously been used to distinguish
glassy structures from liquid ones.
