Mean-field predictions of scaling prefactors match low-dimensional jammed packings

Many results from the mean field theory of glasses and jamming are found to be accurate at low dimensions. While the exponents of scaling power laws between pressure, excess packing fraction, and number of excess contacts are expected to be exact at and above the upper critical dimension of 2, the prefactors to such power laws are not. We present measurements of these prefactors in dimensions 2-10 and show that they do closely follow mean field predictions, suggesting a deeper connection. To explain these findings, we present an exact, first principles derivation for the relation between pressure and excess packing fraction that does not invoke the mean field. These results suggest that mean field theories of critical phenomena may compute more above the upper critical dimension than has been previously appreciated.