and φ$_{g}$ ). This immediately leads to a relation between the typical size of the jumps after time τ and the required scale of the cooperative motion: 6 FIG. 6: Left: Four-point correlator G ∗ $_{4}$( R ) = G 4 ( R ; σ$_{φ}$ ( τ $^{∗}$) , τ $^{∗}$) as a function of R for φ = 0 . 8402 (red), 0 . 8417 (black), 0 . 8426 (blue). Right: Dynamical correlation as a function of the packing fraction; (Inset: Rescaling of log [ G ∗ $_{4}$( R ) /G ∗ $_{4}$(0)] as a function of √ R/ξ 4 for 8 densities around φ J ). From [1]. ξ$_{4}$ ( τ ) ∼ r ( τ ) c ∝ τ $^{ν}$, τ ≤ τ ∗ (11) G$_{6}$ ( R, R $^{′}$, a$^{∗}$, τ $^{∗}$) = 〈 δq t ∗ i δq t ∗ j δq t ∗ k 〉 ∣ t ; | ⃗ R$_{i}$ − ⃗ R$_{j}$ | = R ; | ⃗ R$_{i}$ − ⃗ R$_{k}$ | = R ′ , (12) χ$_{4}$ ( a, τ ) = 1 N 〈( ∑ i δq t i ( a, τ ) ) $^{2}$〉$_{t}$ . (10) This very simple argument predicts that ξ$_{4}$ should be ∼ 10 2 − 10 3 times larger than typical displacements, which is indeed the case (see Fig. 6). Furthermore, using τ ∗ ∝ τ$_{D}$ one finds r ( τ $^{∗}$) ∝ r$_{max}$ , and therefore, using the truncated L´ evy flight model above, a power-law di- vergence of ξ$_{4}$ ( φ ) as | φ − φ$_{J}$ | − $^{ζ}$. This divergence might change if c strongly depends on the distance | φ − φ$_{J}$ | . Turning now to higher cumulants of the distribution of Q ( a $^{∗}$, τ $^{∗}$), one can show that they are related to the space-integral of higher order correlation functions of the dynamical activity. For example, the skewness ς$_{6}$ of Q is 1 /N 2 times the space integral of the 6-point correlation function, defined as: show that χ$_{4}$ ( a, τ ) is related to the variance of Q as (see e.g. [4] for a review): where δq t ∗ i is a shorthand notation for δq t i ( a $^{∗}$, τ $^{∗}$). Similar relations hold for higher moments. The sim- plest scenario is that all these higher order correlation functions are governed by the same dynamical correla- tion length ξ$_{4}$ , extracted from G$_{4}$ ( R, a, τ ). This is what happens in the vicinity of standard phase transitions, for example. If this is the case, and provided that χ ∗ 4 ∝ ξ 2 $_{4}$, one can show that the following scaling relations should hold: We have shown in [1] that χ$_{4}$ ( a, τ ) has, for a given φ , an absolute maximum χ ∗ 4 = χ$_{4}$ ( a $^{∗}$, τ $^{∗}$), which as expected sits on the line corresponding to ¯ Q ( a, τ ) ≈ 1 / 2, i.e. such that half of the particles have moved by more than a . The amplitude of this maximum, which can be interpreted as a number of dynamically correlated grains, grows as φ approached φ$_{J}$ , indicating that the jump motion of the grains becomes more and more collective as one enters the superdiffusive L´ evy regime, as anticipated above. Note in particular that τ ∗ behaves in the same way as τ$_{sD}$ , the time at which the superdiffusion exponent ν reaches its maximum. We furthermore found [1] that the four-point correlation G ∗ $_{4}$( R ) ≡ G$_{4}$ ( R, a $^{∗}$, τ $^{∗}$) is a scaling function of R/ξ$_{4}$ (see inset of Fig. 6-right), where ξ$_{4}$ ( φ ) is the dynamical correlation length such that χ ∗ 4 ∝ ξ 2 $_{4}$. where ς$_{6}$, κ$_{8}$ are respectively the skewness and kurtosis of 1 /N ∑ i q$_{i}$ (related to the 6- and 8-point connected corre- lation functions), and ς$_{c,}$$_{6}$, κ$_{c,}$$_{8}$ the corresponding values at the critical point, i.e. for systems of size N smaller than the correlation volume χ ∗ 4 ∝ ξ 2 $_{4}$. If these scaling relations are valid, the determination of ς$_{c,}$$_{6}$, κ$_{c,}$$_{8}$ using the above equations should give the same values for any φ close to φ$_{J}$ . Unfortunately, our statistics is not suffi- cient to make definitive statements, although the data is indeed compatible with such scalings. The notable fea- ture is that both ς$_{c,}$$_{6}$ ≃ − 1 and κ$_{c,}$$_{8}$ ≃ − 5 are found to be negative , although the error bar on both quantities is large. Interestingly, if we assume that the displace- ments are perfectly correlated within a correlation blob of size ξ$_{4}$ , the L´ evy flight model with µ = 1 . 6 makes the following predictions: All these results were reported in [1] and are re- called here for completeness. The behavior of χ ∗ $_{4}$( φ ) was furthermore shown in [2] to be well accounted for by the following upper bound, derived in [3, 4]: χ$_{4}$ ≥ ( ∂ ¯ Q/∂φ ) $^{2}$〈 φ $^{2}$〉 $_{c}$, where 〈 φ $^{2}$〉 c is the variance of the local density fluctuations. Here, we want to present further speculations, first on the relation between the size of the ‘micro-jumps’ and dynamic correlations, and then on the higher moments of the distribution of Q ( a, τ ). By conservation of the number of particles, the local change of density δφ is related to the divergence of the displacement field by: δφ/φ = ⃗ ∇ · ⃗ r . If we invoke a kind a Reynolds dilatancy criterion whereby the local density must fall below some threshold for the system to move, the displacement field must be correlated over some length ξ$_{4}$ such that ⃗ ∇ · ⃗ r ∼ r/ξ$_{4}$ ≈ c , where c is a small number, possibly dependent on φ , and of the order of 10 − 3 − 10 − 2 ( i.e. the relative difference between φ$_{J}$ in qualitative agreement with our data (note however that there is an unknown proportionality factor in eq. (13). A Gaussian diffusion model, on the other hand, ς$_{c,}$$_{6}$ ≈ − 0 . 12; κ$_{c,}$$_{8}$ ≈ − 1 . 37 , (14) ς$_{6}$ ∼ ς$_{6}$$_{,c}$ √ χ ∗ 4 N , κ$_{8}$ ∼ κ$_{c,}$$_{8}$ χ ∗ 4 N (13)