−2 −1 0 1 2 3 10 −4 10 −2 10 0  z P(  z) 2 3 4 6 8 10 16 24 48 0 10 20 30 40 50 −1.5 −1 −0.5 0 cluster size  z 5 contacts actually depend on the cluster size. To elu- cidate this last point, we compute the distributions of δz = z − 2 n$_{m}$/d within clusters of a given size N (see figure 6a). Apart from discretization effects at small size and the shrinking of the width with ∼ N $_{−}$1 / $^{2}$, the distri- butions progressively shift from negative δz to a slightly positive one. Note also that the minimal value of δz goes from the marginal local criterion for the smallest one (all studied clusters must be composed of locally sta- ble particles) to close to the global stability criterion for the largest one (figure 4b). Altogether these clusters ap- pear to be dynamically selected according to their stabil- ity, which from a local mechanical proprety progressively builds up self consistently with their size towards the generalized isostaticity criterion. The picture that emerges from this analysis is the fol- lowing. The idea that the whole system is isostatic at the avalanche onset is too simple because it ignores the anisotropy and inhomogeneity of the pile, which trans- poses to the critical contacts. Such important effects of the anisotropy have been strongly emphasized be- fore [5, 15] and it is clearly a too strong assumption to elude them [26]. This suggests an even deeper his- tory dependence of frictional piles, which calls for a re- fined description of the texture beyond the introduction of the number of critical contacts. For the isotropically compressed packings of [20, 22], it was also noted that only extremely slowly equilibrated packings unjam at δz gen = 0; for those packings, the critical contacts are indeed randomly distributed [27]. The appearance of system-spanning marginally stable clusters is an intrigu- ing mechanism of unjamming; one which cannot exist in frictionless systems if the packing structure and hence the local contact numbers are to remain homogeneous. This suggests that the frictional and the frictionless jamming transition may be more different than expected. Fur- ther insight in that matter could be gained from new ex- periments with two dimensional packings of photoelastic discs; in particular, the generalized isostaticity diagram could have direct relevance to the cyclic shear experi- ments of [28]. where s is the size of the cluster in particles. We have estimated δ = 2 . 5 ± 0 . 25 from the distribution just above onset (see dashed line) and the inset (a) dis- plays the cluster scale ξ as a function of the distance to avalanche onset. We observe a sharp upturn close to the avalanche, especially in the last 10 frames. Sim- ilarly, the ensemble-averaged size of the largest cluster also grows and presents an upturn just before the onset of the avalanche (see inset (b) of Figure 5). Hence, de- spite their intrinsic marginal stability, and provided that the clusters are indeed quasi unidirectional, which was also established in [15], the maximal size recorded here corresponds to clusters reaching system-spanning size at the onset of the avalanche. onset: FIG. 6: Cluster size effects on stability: a - Distribution of δz for different cluster sizes, with binning that reflects the fun- damental discreteness of the distributions. The curves have been shifted for clarity. b - In black (squares) is 〈 δz 〉 for differ- ent cluster sizes and the global stability criterion (dash-dot). The blue dots are the minimal δz obtained for each cluster size; the dashed line is the marginal stability criterion for a single grain. P ( s ) = 1 s δ e − $^{s/ξ}$, (3) ACKNOWLEDGEMENTS We would like to thank St´ ephanie Deboeuf for having shared her data with us. S. Henkes and C. Brito thank the FOM foundation for funding. O. Dauchot would like to thank the KNAW for the scientific opportunities given by his visiting professorship in Leiden. We acknowledge helpful discussions with Martin van Hecke. In the above discussion, we have assumed that the av- erage contact number inside the clusters is roughly equal to the one for the whole pile. However given the in- crease of the exponential cut-off when approaching the avalanche, one suspects that the average contact num- ber and to a lesser extent the average number of critical