FIG. 2: Left : Typical trajectories of a grain during a 10000 cycle acquisition. The packing fractions are φ = 0 . 8402 (red), 0 . 8417 (orange), 0 . 8422 (yellow), 0 . 8430 (green). φ J would stand between the orange and yellow trajectories. Right : Fraction of broken neighborhood relations (links in the Vorono¨ı tessellation sense), as a function of time for all packing fractions. FIG. 1: Left: Sketch of the experimental setup. Right: Pic- ture of the grains together with an illustration of the relax- ation pattern of the field q i at τ ∗ (see text below for details). regime, accurately given by an isotropic L´ evy stable dis- tribution L$_{µ}$ . Furthermore, the exponent µ characteriz- ing this L´ evy stable distribution is equal to 1 /ν , where ν is the superdiffusion exponent, precisely what one would expect if the motion of the grains was a L´ evy flight , i.e. a sum of uncorrelated individual displacements with a power-law tail distribution of sizes such that the vari- ance of the distribution diverges [5, 6]. This divergence only occurs at φ = φ$_{J}$ , but is truncated away from the critical point, which explains why the motion reverts to normal diffusion at very large times. This finding shows that the rearrangements corresponding to the maximum of dynamical correlations cannot be thought of as large scale currents that remain coherent over a long time scale τ $^{∗}$. Superdiffusion is not induced by long-range temporal correlations of the velocity field, as was surmised in [1]. Quite on the contrary, the total displacement on scale τ ∗ is made of a large number of temporally incoherent jumps with a broad distribution of jump sizes. As is well known, a L´ evy flight is, over any time τ , dominated by a handful of particularly large events. Hence, the only pre- dictability of the total displacement between t = 0 and t = τ ∗ on the basis of the motion in any early period, say between t = 0 → τ $^{∗}$/ 10, comes from the presence of one of these large jumps in [0 , τ $^{∗}$/ 10]. The situation for frictional grains might therefore be quite different from what happens in supercooled liquids or frictionless hard spheres, when soft modes should indeed play an impor- tant role. described in detail in [1], and summarized in Fig. 1. The stroboscopic motion of a set of 8500 brass cylin- ders (“grains”) is recorded by a digital video camera. The cylinders have diameters d$_{1}$ = 4 ± 0 . 01 mm and d$_{2}$ = 5 ± 0 . 01 mm and are laid out on a horizontal glass plate that harmonically oscillates in one direction at a frequency of 10 Hz and with a peak-to-peak amplitude of 10 mm . The cell has width L ≈ 100 d$_{small}$ , and its length can be adjusted by a lateral mobile wall controlled by a µm accuracy translation stage, which allows us to vary the packing fraction of the grains assembly by tiny amounts ( δφ/φ ∼ 5 10 − $^{4}$). The position of the grains is tracked within an accuracy of 2 . 10 − $^{3}$d$_{1}$ . In the following, lengths are measured in d$_{1}$ units and time in cycle units. Fig. 2-left shows the trajectories of a single grain dur- ing the whole experimental window for packing fractions sitting on both sides of the transition φ$_{J}$ . As expected, the typical displacements are much larger at lower pack- ing fractions. The so-called caging dynamics appears clearly on the trajectories obtained at larger packing frac- tions: the grains seem to stay confined around fixed posi- tions for a long time and then hop around from one cage to another over longer time scales. The striking feature of these curves is the remarkably small amplitude of the motion that we alluded to above. Even at the lowest packing fractions, the grains do not move much further than a fraction of a diameter over the total course of one experimental run. Another evidence of the absence of structural relaxation is given by the low percentage of neighboring links (in the Vorono¨ı sense) that are broken during the 10 4 cycles of an experimental run (see Fig. 2- right). This fraction goes from around 5% for the loosest packing fraction to around 0 . 2% for the densest. This confirms that for densities around φ$_{J}$ , the system is deep in the glass phase where structural relaxation is absent, On the other hand, the very fact that individual jumps are broadly distributed in such a dense system where grains can hardly move means that these large jumps are necessarily collective, with a direct relation between the size of the jump and the number of particles involved that we discuss below. The physical connection between su- perdiffusion and dynamical heterogeneities in this system becomes quite transparent. Note however that although we speak about “broad distributions” and “large jumps”, one should bear in mind that all of this takes place on minuscule displacement scales, a few 10 − 2 of the grains diameter! Our L´ evy flights are therefore Lilliputian walks with a diverging second moment... The experimental set-up and the quench protocols are STATISTICAL CHARACTERIZATION OF THE MOTION OF GRAINS 2