  Jamming is a phenomenon shared by a wide variety of systems, such as granular
materials, foams, and glasses in their high density regime. This has motivated
the development of a theoretical framework capable of explaining many of their
static critical properties with a unified approach. However the dynamics
occurring in the vicinity of the jamming point has received little attention
and the problem of finding a connection with the local structure of the
configuration remains unexplored. Here we address this issue by constructing
physically well defined structural variables using the information contained in
the network of contacts of jammed configurations, and then showing that such
variables yield a resilient statistical description of the particle-wise
dynamics near this critical point. Our results are based on extensive numerical
simulations of systems of spherical particles that allow us to statistically
characterize the trajectories of individual particles in terms of their first
two moments. We first demonstrate that, besides displaying a broad distribution
of mobilities, particles may also have preferential directions of motion. Next,
we associate each of these features with a structural variable computed
uniquely in terms of the contact vectors at jamming, obtaining considerably
high statistical correlations. The robustness of our approach is confirmed by
testing two types of dynamical protocols, namely Molecular Dynamics and Monte
Carlo, with different types of interaction. We also provide evidence that the
dynamical regime we study here is dominated by anharmonic effects and therefore
it cannot be described properly in terms of vibrational modes. Finally, we show
that correlations decay slowly and in an interaction-independent fashion,
suggesting a universal rate of information loss.
