the local slope of the stress-strain curve, and also the associated contact numbers z$_{avg}$ . If we take these scaling properties as a signature of the criti- cality of point J, we can conclude that for the range of volume-fractions considered we are in the “criti- cal regime”. As additional characterization of the ensemble of elastic states we report the probability distributions of shear moduli and contact numbers, respectively. Maybe surprisingly all the obtained distributions have approximately the same shape and can be su- perimposed on a single master curve. To achieve this we center each distribution around the average value and rescale the width with a factor p $^{α}$N $^{β}$. By looking carefully at the individual distribu- tions we do observe a slight trend towards the de- velopment of non-Gaussian tails close to φ$_{c}$ . While non-Gaussian distributions are to be expected close Figure 3: (Top) Average nonlinear elastic shear modulus g$_{avg}$ as function of pressure p . (Bottom) Probability distribution P ( g ) centered around av- erage value g$_{avg}$ and rescaled width according to Δ g = p 0 . $^{25}$/N0 . $^{5}$. Black solid line is a Gaussian pdf. Figure 4: (Top) Average contact number z as func- tion of pressure p . The values for z$_{0}$ ≡ z ( p = 0) are determined from a best fit. They are smaller than z$_{c}$ = 4 as to the presence of rattlers, which have not been accounted for. (Bottom) Probability dis- tribution P ( z ) centered around average value z$_{avg}$ and rescaled width according to Δ z = p $_{−}$0 . $^{35}$/N0 . 5 to critical points, 36 the effect is quite small and all distributions have a well developed Gaussian core. The pronounced small- g tail of P ( g ) is due to shear moduli that extend down to zero. Similar tails have been observed in 8 and related to a softening of the response upon approach towards plastic instabil- ities. Indeed, we found that manually suppress- ing states close to plastic events, the weight in the small- g tail is reduced. For the width of the g-distribution we obtain Δ g = p 0 . $^{25}$/N0 . $^{5}$. Thus, the absolute width of the distribution decreases with decreasing pressure, while the relative width, Δ g/g$_{avg}$ diverges at point J. For the contact numbers, on the other hand, we find a divergence of the absolute width itself, Δ z = p $_{−}$0 . $^{35}$/N0 . $^{5}$. These enhanced fluctuations certainly support the view of δz as an order pa- 5 10 -2 10 -1 10 -3 10 -2 10 -1 g p ~x 0.5 N = 2500 N = 1600 N = 900 10 -3 10 -2 10 -1 -9 -6 -3  0  3  6 gP(g) (g-g$_{avg}$)/ g 10 -1 10 0 10 -3 10 -2 10 -1 z-z 0 p$^{~x}$$^{0.5}$ N = 2500 N = 1600 N = 900  0.01  0.1  1 -0.8 -0.6 -0.4 -0.2  0 0.2 0.4 0.6 zP(z) (z-z$_{avg}$)/ z