Super-diffusion around the rigidity transition: L´ evy and the Lilliputians F. Lechenault R. Candelier O. Dauchot J.-P. Bouchaud G. Biroli Capital Fund Management, 6-8 Bd Haussmann, 75009 Paris, France CEA Saclay/SPEC, URA2464, L’Orme des Merisiers, 91 191 Gif-sur-Yvette, France LPS, Ecole Normale Sup´erieure, URA D 1306, 24 rue Lhomond 75005 Paris, France. CEA Saclay/SPCSI, Grp. Complex Systems & Fracture, F-91191 Gif-sur-Yvette, France By analyzing the displacement statistics of an assembly of horizontally vibrated bidisperse fric- tional grains in the vicinity of the jamming transition experimentally studied before [1], we establish that their superdiffusive motion is a genuine L´ evy 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. PACS numbers: As the volume fraction of hard grains is increased be- yond a certain point, the system jams and is able to sus- tain mechanical stresses. This rigidity/jamming transi- tion has recently been experimentally investigated [1, 2] in an assembly of horizontally vibrated bi-disperse hard disks, using a quench protocol that produces very dense configurations, with packing fractions beyond the glass density φ$_{g}$ , such that the structural relaxation time τ$_{α}$ is much larger than the experimental time scale. There is a density range φ$_{g}$ < φ < φ$_{a}$ where the strong vibration can still induce micro-rearrangements through collective contact slips that lead to partial exploration of the por- tion of phase space restricted to a particular frozen struc- ture. For φ$_{g}$ < φ < φ$_{J}$ ≈ 0 . 842, the system is frozen but not rigid; the system can only sustain an external stress for φ larger than φ$_{J}$ , which appears as a genuine critical point where a dynamical correlation length and a cor- relation time simultaneously diverge, showing that the dynamics occurs by involving progressively more collec- tive rearrangements. One of the most surprising result of [1] was the dis- covery of a superdiffusive regime in the vicinity of φ$_{J}$ . In a time range that diverges when φ → φ$_{J}$ , the typical displacement of the grains grows as τ $^{ν}$, with ν > 1 / 2, arXiv:1001.1765v1 [cond-mat.soft] 11 Jan 2010 i.e. faster than the familiar diffusive √$_{τ}$ law, while al- ways remaining small compared to the diameter of the grains! It was also found that dynamical heterogeneities reach a maximum when τ = τ $^{∗}$( φ ), precisely when the superdiffusive character of the motion is strongest [2], in agreement with the general bound on the dynamical sus- ceptibility established in [3, 4]. This superdiffusion was interpreted as the existence of large scale convective cur- rents, which were tentatively associated to the extended soft modes that appear when the system loses or acquires rigidity at φ$_{J}$ [7], and along which the system should fail. A similar interpretation of the dynamics of partic- ulate systems close to the glass or jamming transitions was promoted in [8]. The intuitive idea is that energy barriers that the system has to cross to equilibrate are essentially in the directions (in phase space) defined by the soft modes. Within this picture, the harmonic motion of the particles around a metastable configurations can be used to guess the structure of the anharmonic barrier crossing events. This is certainly reasonable when the barriers are small, so that the top of the barrier is still in a quasi-harmonic regime. The primary aim of the present paper is to revisit the above interpretation for our system of hard-disks with friction, in the light of a deeper analysis of the statistics of displacements. To our surprise, we find that the distri- bution of rescaled displacements is, in the superdiffusive INTRODUCTION CEA Saclay/IPhT, UMR2306, L’Orme des Merisiers, 91 191 Gif-sur-Yvette, France (Dated: June 18, 2018)