predicts ς$_{c,}$$_{6}$ = 0 and κ$_{c,}$$_{8}$ = − 6 / 5, corresponding to a uniform distribution of Q in [0 , 1]. A kurtosis smaller than − 6 / 5 means that the distribution of Q tends to be spiked around 0 and 1. Finally, as φ increases beyond φ$_{J}$ , the negative skewness of Q markedly increases. This is a sign that the dynamics becomes more and more inter- mittent, “activated”, with a few rare events decorrelat- ing the system completely, while most events decorrelate only weakly. Since we fix ¯ Q ( a $^{∗}$, τ $^{∗}$) = 1 / 2, this leads to a diverging negative skewness in the limit where the proba- bility of rare decorrelating events tends to zero. The idea of characterizing the skewness of Q might actually be an interesting tool to characterize the strength of activated events in other glassy systems. 

The central finding of this work is that the superdiffu- sive motion of our frictional grains in the vicinity of the jamming transition appears to be a genuine L´ evy flight, but with ‘jumps’ taking place on a Lilliputian scale. This interpretation contrasts with the original suggestion made in [1], where anomalous diffusion was ascribed to persistent, spatially correlated currents. The vibration of the plate induces a broad distribution of jumps that are random in time, but correlated in space, and that can be interpreted as micro-crack events on all scales. As the volume fraction departs from the critical jam- ming density φ$_{J}$ , this distribution of jumps is truncated at smaller and smaller jump sizes, inducing a crossover towards standard diffusive motion at long times. This picture severely undermines the usefulness of harmonic modes as a way to rationalize the dynamics of our sys- tem (although this conclusion might of course not carry over to frictionless grains, or to thermal systems). The detailed study of these modes, and the difficulty to ana- lyze them in the present system, is discussed in [15]. 

We have also presented several other speculations about the relation between the dynamical correlation length ξ$_{4}$ and the size of the jumps, and the structure of higher order dynamical cumulants. The idea of using the 6-point skewness as a quantitative measure of the importance of activated events in the dynamics of glassy systems seems to us worth pursuing further. 

Note added and acknowledgment: We want to thank L. Berthier for interesting discussions about a preprint of his and collaborators [16] (that appeared in the very last stages of the present work), and where effects simi- lar to those discussed here are observed numerically in a 

## CONCLUSION 

model system. We thank him in particular for insisting on the importance of measuring velocity correlations in our system. 

[1] F. Lechenault, O. Dauchot, G. Biroli, and J.-P. Bouchaud, Europhys. Lett. 83 , 46003 (2008). 

[2] F. Lechenault, O. Dauchot, G. Biroli, and J.-P. Bouchaud, Europhys. Lett. 83 , 46002 (2008). 

[3] L. Berthier, G. Biroli, J.-P. Bouchaud, L. Cipelletti, D. El Masri, D. L’Hote, F. Ladieu and M. Pierno, Science 310 , 1797 (2005) 

[4] L. Berthier, G. Biroli, J.-P. Bouchaud, W. Kob, K. Miyazaki and D. Reichman, J. Chem. Phys. 126 184503 (2007). 

[5] G. Zaslavsky, U. Frisch, M. Shlesinger (Edts.), L´ evy Flights and Related Topics in Physics , Lecture Notes in Physics 450 , Springer, New York (1995). 

[6] J.-P. Bouchaud, A. Georges, Anomalous diffusion in dis- ordered media: statistical mechanisms, models and phys- ical applications , Phys. Rep. 195 , 127 (1990). 

[7] M. Wyart, S. R. Nagel and T. A. Witten, Europhys. Lett. , 72 , 486 (2005); M. Wyart, L. E. Silbert, S. R. Nagel, T. A. Witten, Phys. Rev. E 72 051306 (2005); C. Brito, M. Wyart, Europhys. Lett. 76 149 (2006). 

[8] A. Widmer-Cooper, H. Perry, P. Harrowell, D. R. Re- ichman, Localised soft modes and the supercooled liq- uid’s irreversible passage through its configuration space , arXiv:0907.0222 

[9] G. Marty, O. Dauchot Subdiffusion and Cage Effect in a Sheared Granular Material , Phys. Rev. Lett. 94 , 015701 (2005). O. Dauchot, G. Marty, G. Biroli Dynamical Het- erogeneity Close to the Jamming Transition in a Sheared Granular Material , Phys. Rev. Lett. 95 265701 (2005). 

[10] A. S. Keys, A. R. Abate, S. C. Glotzer, D. J. Durian, Nature Physics 3 , 260 (2007). 

[11] R. Candelier, O. Dauchot, G. Biroli Phys Rev. Lett. 102 , 088001 (2009) 

[12] G. Samorodnitsky, M. S. Taqqu, Stable Non-Gaussian Random Processes , Chapman & Hall, New York, 1994. 

[13] I. Koponen, Phys. Rev. E 52 , 1197 (1995). 

[14] J.-F. Muzy, J. Delour, E. Bacry, Eur. Phys. J. B 17 , 537 (2000); E. Bacry, J. Delour and J.F. Muzy, Phys. Rev. E 64 , 026103 (2001). 

[15] C. Brito, O. Dauchot, G. Biroli, J.-P. Bouchaud, com- panion paper. 

[16] C. Heussinger, L. Berthier, J.-L. Barrat, Superdiffu- sive, heterogeneous, and collective particle motion near the jamming transition in athermal disordered materials , arXiv:1001.0914 

[17] Technically, however, these logarithmic correlations sug- gest that our superdiffusion process is a multifractal L´ evy flight , by analogy with the multifractal random walk in- troduced in [14]. We will not dwell on this interesting new process, that would deserve a theoretical study of its own. 

