0 0.1 0.2 0.3 0.2 0.4 0.6 N$_{c}$/N N c /N −15 −10 −5 0 5 10 15 0.15 0.25 0.35 0.45 C( ) 3 off of a maximum of 2% of contacts lost between frames to determine the end of the quiet period (in red). In Figure 2(c), this criterion has been applied to a sample configuration, and it is clearly able to capture the loca- tion of significant events in the packing. We have checked the robustness of this criterion by varying the cutoff be- tween 1% and 4%, which typically changes the location of the onset of the avalanche by 1-2 frames, correspond- ing to an error in θ$_{m}$ of 0 . 05 ◦ − 0 . 1 ◦ . We have observed that the position of the avalanche onset is very variable, and that there may be several avalanches of various sizes within a single run. In the following all ensemble aver- ages are performed as a function of θ$_{m}$ − θ , where θ$_{m}$ is the location of the next avalanche onset. Figure 2d shows the probability density of the system in the parameter space ( z, n$_{m}$ ), together with the trace of one sample run. The black (respectively the red) symbols correspond to the quiet (resp. avalanche) period. Two features are apparent. First, the quiet period and the avalanche occupy very well separated regions of the pa- rameter space and the transition between them is abrupt. During the quiet period, the mean contact number z is quenched, and all of the dynamics is due to fully mobi- lized contacts appearing and disappearing in the packing (corresponding to an up-and-down motion in the graph). This is confirmed by our observation that throughout the pre-avalanche period, particles move typically less than 1% of their diameter and remain “caged”. When the system enters the avalanche period, the structure of the packing breaks down and the trace moves along a diag- onal. Second, the system crosses the marginal stability line only after the start of the avalanche. This means that the generalized isostaticity criterion, (Eq. 1), is a necessary criterion for stability, but not a sufficient one. FIG. 3: (Color online) Top: Contact properties close to the avalanche onset, θ m − θ ≈ 0 . 3 ◦ ; clusters of particles with at least one critical contact (in green) are in black. Particles with failing contacts belonging to a cluster are red, failure of the other particles is in blue. The red arrow indcates the direction of gravity – note the directionality of the clusters. Bottom: Spatial (left) and temporal (right) correlation between clus- ters and failure during the quiet period. Left: Fraction of failure which occurs in a cluster, N f c /N c as a function of frac- tion of the system in a cluster, N c /N . The dashed line is the limit where failure and clusters are uncorrelated. Right: Temporal correlation between the maximum cluster size at time t and the number of particles that fail between t and t + τ ; the units are frames each corresponding to Δ θ = 0 . 1 ◦ . Generalized isostaticity is a global, mean-field type cri- terion, and it is likely to overestimate the stability of the packing if the system behaves in an heterogeneous way. We have checked that such heterogeneities are not related to the geometry. If we exclude the top and bottom sin- gle layers of particles, which cannot easily be integrated into the counting, no part of the system is critical at the start of the avalanche (note that we only used the bulk of the system for Figure 2d). We also have cut the system both into lanes, and several other patterns and found that the same general conclusion holds: within a scale of a few frames, the avalanche onset measured by this method is the same for all sub-parts of the system ensuring that the avalanche is truly a global property of the system (see also Figure 4b). Note that for a granu- lar bed much deeper than our relatively shallow layer of particles, one may observe a stronger dependance of n$_{m}$ on depth than in the present case and that the system could then eventually separate. Since the contact number is quenched during the quiet period and all changes in δz gen are borne by the criti- cal contacts, while the remainder of the system does not contribute, we can divide the system into two subpopu- lations: those particles with at least one fully mobilized contact, and those without. Figure 3, top, illustrates the critical contacts close to the onset of the avalanche. It be- comes apparent that the particles with critical contacts organize in rapidly fluctuating clusters . These clusters are rather anisotropic and their preferential direction of alignment makes an angle of roughly 60 ◦ with respect to gravity. This is in agreement to the Mohr-Coulomb fail- ure criterion and its extension in the context of elasto- plasticity [25], which would predict a failure angle of 1 / 2 ( π/ 2 − atan( µ )) ≃ 30 ◦ , that is along the slip plane at the critical contacts, normal to the contact orienta- tion. We now turn to the relation between these clusters and local failure, as measured by the contacts lost between two consecutive frames. We find both spatial and tempo- Inspired by Staron’s work [13], we identify the crit- ical contacts as a natural candidate for the source of the heterogeneity suggested by the above observations.