Marginality gap in Mari-Kurchan mean-field model for jammed packings

A marginal system is one for which there exists an infinitesimal deformation that will lead to instability. Frictionless sphere packings at jamming, as well as infinite packings above jamming, seem to meet this definition. Further, the scaling of the excess contact number with pressure is consistent with the mean-field expectation of marginal stability. However, the prefactor in three-dimensional systems is slightly larger than predicted. This discrepancy in the prefactor, termed the “marginality gap”, is expected to vanish in the mean-field limit. We investigate numerically the Mari-Kurchan model of jammed packings, in which a Gaussian random shift is added to each separation of pairs of particles in the pair potential. By tuning the amplitude of the random shift, we study the range from jammed packings to the mean-field limit to see how the marginality gap evolves.