  We present an analysis of finite-size effects in jammed packings of N soft,
frictionless spheres at zero temperature. There is a 1/N correction to the
discrete jump in the contact number at the transition so that jammed packings
exist only above isostaticity. As a result, the canonical power-law scalings of
the contact number and elastic moduli break down at low pressure. These
quantities exhibit scaling collapse with a non-trivial scaling function,
demonstrating that the jamming transition can be considered a phase transition.
Scaling is achieved as a function of N in both 2 and 3 dimensions, indicating
an upper critical dimension of 2.
