3 sented in Fig. 3, see [1]. Note the very small values of σ$_{φ}$ ( τ ) ≪ d$_{1}$ at all timescales, which is a further indica- tion that the packing indeed remains in a given struc- tural arrangement. At low packing fractions φ < φ$_{J}$ and at small τ , σ$_{φ}$ ( τ ) displays a sub-diffusive behavior. At longer time, diffusive motion is recovered. As the packing fraction is increased, the typical lag at which this cross- over occurs becomes larger and, at first sight, σ$_{φ}$ ( τ ) does not seem to exhibit any special feature for φ = φ$_{J}$ (corre- sponding to the bold line in Fig.3). Above φ$_{J}$ , an inter- mediate plateau appears before diffusion resumes. How- ever, a closer inspection of σ$_{φ}$ ( τ ) reveals an intriguing behavior, that appears more clearly on the local logarith- mic slope ν = ∂ ln σ$_{φ}$ ( τ ) /∂ ln( τ ) shown in Fig. 3. When ν = 1 $_{2}$, the motion is diffusive, whereas at small times, ν < 1 2 indicates sub-diffusive behavior. At intermedi- ate packing fractions, instead of reaching 1 2 from below, ν overshoots and reaches values ≈ 0 . 65 before reverting to 1 2 from above at long times. Physically, this means that after the sub-diffusive regime commonly observed in glassy systems, the particles become super-diffusive at intermediate times before eventually entering the long time diffusive regime. At higher packing fractions, this unusual intermediate superdiffusion disappears: one only observes the standard crossover between a plateau regime at early times and diffusion at long times. In order to characterize these different regimes, we define three characteristic times: τ$_{1}$ ( φ ) as the lag at which ν ( τ ) first reaches 1 / 2, corresponding to the beginning of the super- diffusive regime, τ$_{sD}$ ( φ ) when ν ( τ ) reaches a maximum ν $^{∗}$( φ ) (peak of super-diffusive regime), and τ$_{D}$ ( φ ) where ν ( τ ) has an inflection point, beyond which the system ap- proaches the diffusive regime. These characteristic time scales are plotted as a function of the packing fraction in the bottom-left panel of Fig. 3. Whereas τ$_{1}$ does not ex- hibit any special feature across φ$_{J}$ , both τ$_{sD}$ and τ$_{D}$ are strongly peaked at φ$_{J}$ . We have also shown on the same graph the time scale τ ∗ where dynamical heterogeneities are strongest (see [1] and below). As noted in the intro- duction, τ ∗ and τ$_{sD}$ are very close to one another and we will identify these two time scales in the following. i.e. in a very different regime from the one studied in [9–11]. We will denote by R t i = ( x t $_{i}$, y t $_{i}$) the vector position of grain i at time t in the center-of-mass frame. Lagged dis- placements are defined as r t $_{i}$( τ ) = R t + τ i − R t i for grain i between time t and t + τ . We have checked in details that the statistics of r t i along the x and the y axes are identi- cal, or more precisely that the motion is isotropic, in spite of the strongly anisotropic nature of the external drive. This in itself is a non trivial observation, that shows that the random structure of the packing is enough to convert a directional large scale forcing into an isotropic noise on small scales. From now on, we will thus focus on the total displacement r t i = | r t i | . We measure typical dis- placements for a density φ and lag τ as the mean absolute displacement, defined as: FIG. 3: Top: Average displacement σ$_{φ}$ ( τ ) ≡ 〈| r t $_{i}$| 〉 i,t (left); and local slope ν = ∂ ln σ$_{φ}$ ( τ ) /∂ ln τ (right) as functions of the lag τ for all studied values of φ . The dotted dark line corresponds to φ J . Bottom left: Characteristic time scales extracted from the analysis of ν ( τ ) including τ $^{∗}$, which will be introduced and discussed in the section devoted to dynamical heterogeneities. Bottom left: Pdf of the displacements r i , P τ ( r ), normalized by σ$_{φ}$ ( τ ), for different values of the lag time τ at φ = 0 . 8446 (right). A Gaussian distribution (black dotted curve) is shown for comparison. We now turn to the probability distribution of the dis- placements r t i ( τ ). We have represented on the bottom- right panel of Fig. 3 the distributions accumulated for all grains and instants for the packing fractions closest to φ$_{J}$ and several values of lag-time τ . The horizontal axis has been normalized by the root mean square dis- placements, and a unit Gaussian is also plotted for com- parison. We find that the tails of the distributions are much fatter than the Gaussian, unveiling the existence of extremely large displacements (compared to typical val- ues) in the region of time lags corresponding to superdif- fusion. As we will discuss below, these fat tails tend to disappear when one leaves the superdiffusion regime, i.e. when | φ − φ$_{c}$ | and/or | ln τ /τ $^{∗}$| become large. Before characterizing more precisely these probability where the time average 〈·〉 t is performed over 10 , 000 cy- cles. We have chosen to estimate σ$_{φ}$ ( τ ) by the mean absolute value deviation instead of the root mean square displacement because – as we shall see – the later diverges when approaching the transition. The evolution of σ$_{φ}$ ( τ ) with the packing fraction is pre- σ$_{φ}$ ( τ ) ≡ 〈 | r t $_{i}$( τ ) | 〉 i,t (1)