4 distributions, we would like to mention that we have re- moved “rattling events” from our analysis, i.e. events where particles make an anomalously large back and forth motion, of amplitude larger than (say) 0 . 1 d$_{1}$ . Some of these events are due to the same grain rattling during a large fraction of the experimental run, while other events are localized in time for a given grain, and could actually well be part of the extreme tail of the distribution seen in Fig. 3. Our point here is that the observed fat-tails are not an artifact due to a few “loose” grains – each and everyone of the grains seem to contribute at one point or another in its history to these fat tails. A neat way to characterize the probability distribution of the displacements is to study the generating function of the squared displacements: FIG. 4: Top: F τ ( x ) as a function of − log$_{10}$( x ) for τ = τ ∗ = 373 (dots, left panel) and 1 / F τ ( x ) − 1 as a function of x for different values of τ (dots, right panel), from 1 (blue) to 10 4 (red). For both plots: φ = φ J and the solid line corresponds to the Gaussian fit 1 / (1 + x ). Bottom-left: Fit of F τ ( x ) for φ ≈ φ J and τ ≈ τ ∗ using a L´ evy distribution model F ( λ, τ )$_{L}$ with index µ = 1 . 6, and comparison with a Student distribution model F ( λ, τ )$_{L}$ with the same small x behavior. Bottom-right: Fitting parameter ε = 1 − µ/ 2 as a function of φ and τ . F ( λ, τ ) ≡ 〈 e − λr t 2 i ( τ $^{)}$〉 i,t = ∞ ∑ k =0 ( − λ ) k k ! 〈 r t 2 k i ( τ ) 〉 . (2) F ( λ, τ )$_{G}$ = ˜ F $_{G}$( x ) = 1 1 + x , (3) As a benchmark, one can easily compute F ( λ, τ ) in the case of isotropic Gaussian diffusion. One obtains: where x = 2 σ$_{φ}$ ( τ ) $^{2}$λ is the scaled variable. When com- puting F ( λ, τ ) for the empirical data, one finds, as ex- pected, systematic discrepancies with the F ( λ, τ )$_{G}$. The rescaling of the different curves when plotted as a func- tion of x , on the other hand, works very well, as can be inferred from the scaling of the probability distributions themselves. the fit suggests that the distribution of displacements is indeed an isotropic L´ evy stable distribution L$_{µ}$ of index µ . We have checked that other distribution functions with power-law tails achieve a much poorer fit to the data. For example, a Student distribution of displacements (which cannot be obtained as the sum of independent jumps, contrarily to the L´ evy distribution) leads to: The most important finding is that the small x be- havior of the empirical F ( λ, τ ) is singular: as shown in the top-right panel of Fig. 4, 1 / F ( λ, τ ) − 1 behaves as x α when x ≪ 1, with α ≈ 0 . 80. It is easy to show that such a singular behavior is tantamount to the ex- istence of a power-law tail in the distribution of r ( τ ), P$_{τ}$ ( r ) ∼ r →∞ r − 1 − µ with α = min(1 , µ/ 2). The value of α therefore corresponds to µ ≈ 1 . 6 < 2, which means that the variance of the distribution of r is formally di- vergent, or at least dominated by a large physical cut- off r$_{max}$ . This suggests to fit F ( λ, τ ) over the whole x regime by the appropriate Laplace transform of a L´ evy stable distribution L$_{µ}$ , that reads for the present isotropic two-dimensional problem [12] : where µ < 2 and C ′ is another numerical constant. Al- though F ( λ, τ )$_{S}$ has the same small x singularity and the same large x behavior as F ( λ, τ )$_{L}$, it turns out not to be possible to adjust the parameter C ′ in such a way to adjust simultaneously the small and large x behavior of the empirical F ( λ, τ ). What is most significant, however, is that the value of the L´ evy index µ corresponding to the best fit turns out to be very close to the one expected in the case of a L´ evy flight where the diffusion exponent ν is given by 1 /µ , since we find ν ≈ 0 . 65 (see Fig. 3) whereas 1 /µ ≈ 0 . 625. Fig. 4 bottom-right gives the value of the fitting pa- where µ < 2 and C is a numerical constant. It is easy to check that for small x , ˜ F $_{L}$( x ) ≈ 1 − C Γ(1+ µ $_{2}$) x µ/ $^{2}$. Our main result is that this function is a very good fit to the empirical data corresponding to φ ≈ φ$_{J}$ and τ ≈ τ $^{∗}$, for all values of x , see the bottom-left panel in Fig. 4. The optimal value of µ is slightly smaller, µ ≈ 1 . 6, than the one obtained by a fit of the small x region. The quality of F ( λ, τ )$_{S}$ = ˜ F $_{S}$( x ) (5) = 1 − ( C $^{′}$x ) µ/ 2 ∫ ∞ 0 d z e − z ( z + C $^{′}$x ) µ/ $_{2}$, F ( λ, τ )$_{L}$ = ˜ F $_{L}$( x ) = ∫ ∞ 0 d z e − z − C ( xz ) µ/ 2 (4)