  No known analytic framework precisely explains all the phenomena observed in
jamming. The replica theory for glass and jamming is a mean field theory which
attempts to do so by working in the limit of infinite dimensions, such that
correlations between neighbors are negligible. As such, results from this mean
field theory are not guaranteed to be observed in finite dimensions. However,
many results in mean field for jamming have been shown to be exact or nearly
exact in low dimensions. This suggests that the infinite dimensional limit is
not necessary to obtain these results. In this paper, we perform precision
measurements of jamming scaling relationships between pressure, excess packing
fraction, and number of excess contacts from dimensions 2-10 in order to
extract the prefactors to these scalings. While these prefactors should be
highly sensitive to finite dimensional corrections, we find the mean field
predictions for these prefactors to be exact in low dimensions. Thus the mean
field approximation is not necessary for deriving these prefactors. We present
an exact, first principles derivation for one, leaving the other as an open
question.
