FIG. 5: Correlation of the instantaneous ( τ = 1) velocity vectors |〈 r t $_{i}$. r t + t ′ i 〉 i,t | (blue circles) versus the correlation of their amplitudes 〈 r t $_{i}$.r t + t ′ i 〉 i,t (red squares) as functions of the lag t $^{′}$, for φ ≈ φ J . 5 rameter ε = 1 − µ/ 2 in the plane φ, τ , obtained from the small x behavior of F ( λ, τ ), corresponding to large dis- placements. We see that the effective values of ε (resp. µ ) become closer to 0 (resp. 2), corresponding to normal diffusion, as when | φ − φ$_{c}$ | and/or | ln τ /τ $^{∗}$| increase. We believe that this corresponds to a so-called “truncated” L´ evy flight [13], with a cut-off value in the distribution of elementary jump size that becomes smaller and smaller as one departs from the critical point, rather than a con- tinuously varying exponent µ . In other words, a plausible scenario that explains the behavior of ε as a function of φ, τ is that the tail of the elementary jump size distribu- tion is given by: where the exponent µ ≈ 1 . 6 is independent of φ , whereas the typical scale of the jumps r$_{0}$ ( φ ) and the cut-off r$_{max}$ ( φ ) both depend on φ . It is reasonable that r$_{max}$ ( φ ) only diverges at φ = φ$_{J}$ , while r$_{0}$ ( φ ) has a regular, de- creasing behavior as a function of φ . In fact, the parame- ter C appearing in the L´ evy distribution above is directly proportional to r µ $_{0}$. The above analysis focused on single grain statistics and established that the displacement of a grain has a L´ evy stable distribution in the superdiffusive regime. This means that the motion of each grain can be de- composed in a succession of power-law distributed jumps. But if “large” jumps can occur in such a jammed system, this necessarily implies that these jumps are correlated in space. In order to characterize these dynamical cor- relations, we have introduced in [1] the following local correlation function: P$_{1}$ ( r ) ∼ [ r$_{0}$ ( φ )] µ r 1+ µ , (6) DYNAMICAL CORRELATIONS Assuming that the jumps are independent, the distri- bution of displacements P$_{τ}$ ( r ) on a time scale τ is given by the τ -th convolution of P$_{1}$ ( r ), which converges to a L´ evy distribution of order µ when r$_{max}$ = ∞ and pre- dicts r ∝ τ ν with ν = 1 /µ (see e.g. [6]). For finite r$_{max}$ , P$_{τ}$ will be very close to L$_{µ}$ in the intermediate regime 1 ≪ τ ≪ τ$_{D}$ , where τ$_{D}$ ∼ ( r$_{max}$/r$_{0}$ ) $^{µ}$, before crossing over to a diffusive regime at very long times. If r$_{max}$ ( φ ) behaves as | φ − φ$_{J}$ | − $^{ζ}$, the diffusion time τ$_{D}$ should di- verge as | φ − φ$_{J}$ | − ζµ when approaching φ$_{J}$ . In order to check directly whether the jumps are in- deed independent, we have measured the correlation of instantaneous ( τ = 1) velocities 〈 r t $_{i}$. r t + t ′ i 〉 $_{i,t}$. As shown in Fig. 5, this correlation function indeed decays ex- tremely fast with the lag t $^{′}$, excluding that superdiffusion could be due to long-range correlations in the displace- ments. However, we discovered that the amplitude of the displacements, proportional to the parameter r$_{0}$ , re- veal long-range correlations, decaying approximately as − ln t $^{′}$. This suggests that some slow evolution takes place during an experimental run, that affects the value of r$_{0}$ over time. A natural mechanism is through the fluctuations of the local density φ (see below), that feed- back on r$_{0}$ . The approximately logarithmic decay of the correlations is interesting in itself and characteristic of multi-time scale glassy relaxation. These long-ranged correlations slow down but fortunately do not jeopardize the convergence towards a L´ evy stable distribution.[17] Therefore, the above interpretation in terms of a jump dominated superdiffusion, instead of persistent currents, is warranted. where δq t i ( a, τ ) = q t i ( a, τ ) − ¯ Q ( a, τ ) and is plotted for three packing fractions on fig. (6)-left. Its sum over all R ’s defines the so-called dynamical susceptibil- ity χ$_{4}$ ( a, τ ). Through simple manipulations, it is easy to G$_{4}$ ( R, a, τ ) = 〈 δq t i ( a, τ ) δq t j ( a, τ ) 〉 ∣ t ; | ⃗ R$_{i}$ − ⃗ R$_{j}$ | = R (9) Note that it is closely related to the previously introduced generating function through ¯ Q ( a, τ ) = F ( λ = 1 2 a $_{2}$, τ ) , which we have characterized in the previous section. The dynamical (four-point) correlation function is de- fined as the spatial correlation of the q field: where a is a variable length scale over which we probe the motion. Essentially, q t i ( a, τ ) = 0 if within the time lag τ the particle has moved more than a , and q t i ( a, τ ) = 1 oth- erwise: q t i ( a, τ ) is a mobility indicator. The average over all i and t defines a function akin to the self-intermediate scattering function in liquids: ¯ Q ( a, τ ) ≡ 〈 1 N ∑ i q t i ( a, τ ) 〉 $_{t}$. (8) q t i ( a, τ ) ≡ exp − r t 2 i ( τ ) 2 a 2 , (7)