  By analyzing the displacement statistics of an assembly of horizontally
vibrated bidisperse frictional grains in the vicinity of the jamming transition
experimentally studied before, we establish that their superdiffusive motion is
a genuine Levy flight, but with `jump' size very small compared to the diameter
of the grains. The vibration induces a broad distribution of jumps that are
random in time, but correlated in space, and that can be interpreted as
micro-crack events at all scales. As the volume fraction departs from the
critical jamming density, this distribution is truncated at a smaller and
smaller jump size, inducing a crossover towards standard diffusive motion at
long times. This interpretation contrasts with the idea of temporally
persistent, spatially correlated currents and raises new issues regarding the
analysis of the dynamics in terms of vibrational modes.
