Two diverging length scales in the structure of jammed packings

At densities higher than the jamming transition for athermal, frictionless repulsive spheres we find two distinct length scales, both of which diverge as a power law as the transition is approached. The first, ${\xi }_Z$, is associated with the two-point correlation function for the number of contacts on two particles as a function of the particle separation. The second, ${\xi }_f$, is associated with contact-number fluctuations in subsystems of different sizes. On scales below ${\xi }_f$ the fluctuations are highly suppressed, similar to the phenomenon of hyperuniformity usually associated with density fluctuations. The exponents for the divergence of ${\xi }_Z$ and ${\xi }_f$ are different and appear to be different in two and three dimensions.