High Dimensional Scaling Prefactors

In critical phenomena, quantities generally obey power law relations from a critical point. In jamming, the exponents to the power laws between pressure, packing fraction, and number of excess contacts can be derived exactly in the limit of infinite dimensions by the mean field theory of glasses and jamming. While these exponents are expected to be exact above the upper critical dimension, the prefactors to these power law scalings are generally not. We computationally examine these prefactors in up to 10 dimensions and find that they do follow expected behavior from mean field theory. We additionally explore quantitative methods of explaining these prefactors that do not invoke the mean field assumption.