Figure 6: (Top) Distribution of elastic branch length normalized with average values Δ γ$_{avg}$ . (Bot- tom) Scaling of Δ γ$_{avg}$ with pressure p . The ratio g = Δ σ$_{avg}$/ Δ γ$_{avg}$ , which defines a shear-modulus, is consistent with the scaling in Fig. 3. The authors have argued for a universal value of the “flow-strain” σ$_{y}$/g of a few percent. Appar- ently, this can only be true far away from φ$_{c}$ . As the yield-stress vanishes faster than the shear mod- ulus, one finds a ratio σ$_{y}$/g ∼ δφ 1 / 2 that vanishes at point J. Thus, particle configurations at the on- set of jamming are highly fragile and susceptible to even minute changes in the boundary conditions. The average stress-drop as well as the average length of elastic branches change with pressure and system-size as displayed in Figs. 5 and 6 $^{2}$. As a function of pressure one observes an increase but with a slope that depends on system-size. The av- $^{2}$Note, that these values also depend on the elementary strain step used in the simulations. For any finite step-size there will be some small plastic events, that cannot be re- solved but only lead to an apparent reduction of the stress increase. This will artificially increase the length of elastic branches and decrease the weight of the small Δ σ -tail of the stress-drop distribution. 3 The effective exponents range from 0 . 8 to 0 . 9 as com- pared to an exponent 0 . 95 for the yield-stress $^{4}$If we assume for the isostaticity length l ⋆ = 1 /δz , we would have values l ⋆ ≈ 5 at φ = 0 . 846 and l ⋆ ≈ 1 at φ = 0 . 9 ( z -values taken from Fig. 4). $^{5}$To calculate the overlap function we use the non-affine displacements in the gradient direction (irrespective of the distance to the wall). Let us start by presenting the results from the quasistatic simulations. To define a dynamical cor- relation length we study heterogeneities in the par- ticle mobilities. To this end we use the overlap- function 43,44 Let us now turn to the dynamics of the system. In particular we want to characterize dynamic corre- lations in the motion of particles. While at volume- fractions above φ$_{c}$ the isostaticity length-scale l ⋆ is clearly finite $^{4}$, there is nevertheless a large dynam- ical length-scale related to the flow arrest. This has, for example, been evidenced in a system of Lennard-Jones particles with dissipative dynam- ics. 15 We will show below that a similar length- scale occurs in our system of purely repulsively in- teracting particles, independent of the distance to φ$_{c}$ . 3.2 Dynamical correlations erage stress-drops increase somewhat slower with pressure than the yield-stress $^{3}$. The relative stress fluctuations Δ σ$_{avg}$/σ$_{y}$ are thus slightly enhanced at small pressures close to φ$_{c}$ . The same trend is also visible in the total stress fluctuations as calculated by 〈 ( σ − 〈 σ 〉 ) 2 〉 . The overall scale of both, Δ σ$_{avg}$ and Δ γ$_{avg}$ , de- creases with system-size to give a smooth stress- strain relation in the thermodynamic limit. Pre- vious studies 8,10,11 have observed a scaling of the stress-drops with N $_{−}$1 / $^{2}$. This includes 8 a sys- tem of harmonically interacting particles, as stud- ied here, but at a rather high pressure. In gen- eral, we observe a weaker dependence on system- size, with an effective exponent that increases with pressure. Our data is consistent, however, with the value of 1 / 2 being the relevant high-pressure limit. 10 -3 10 -2 10 -1 10 0  0  1  2  3  4  5 avg P avg 4 10 -4 10 -3 2 10 -3 10 -3 10 -2 avg p N = 900 = 1600 = 2500 of particles undergoing nonaffine displacements u$_{i}$$_{na}$ during a strain interval of γ $^{5}$. Particles mov- ing farther than the distance a (“mobile”), have 〈 Q ( γ, a ) 〉 = 〈 1 N N ∑ i =1 exp [ − u$_{i}$$_{na}$ ( γ ) 2 2 a 2 ] 〉 , (3) 7