10 0 10 1 10 2 10 3 10 4 t -4 10 -3 10 -2 10 -1 Q(t) 10 0 10 1 10 2 10 3 10 4 t FIG. 2: Solid line: Time dependence of the nonlinear contribution in two discretizatio n schemes [Eq. (10) (left frame) and Eq. (53) (right frame)]. We also depict the Galilean contribution (circles) and the excess of fluctuations (diamonds) as defined e.g. in Eq. (51). For both cases, the excess of fluctuations are comparable with the Galilean contribution. In order to analyze the excess of fluctuations that such discretization schemes present with respect to the standard one, we extract the Galilean invariant fluctuations from the quadratic term of the equation of motion. In Fig. 2 we depict the time dependence of the different nonlinear contributions for both alternative discretization schemes. On the left we have the comparison between the discretization scheme Eq. (47) and the one in Eq. (48), while on the right we compare the scheme in Eq. (47) to the one in Eqs. (30,31). The first point to note is the rapid saturation of nonlinearities. This behavior is consistent with the rapid saturation of local fluctuations, which behave as G ( ℓ, t ) ∼ t 2 β and saturate for t ≫ ℓ z [35]. We also observe that for both cases, the excess of fluctuations are comparable to (or even larger than) the Galilean contribution. As we pointed out in previous sections, this result, together with the persistence of KPZ scaling shown in Fig. 1, would imply that Galilean invariance is not such a fundamental symmetry as usually considered. It is worth remarking that the present results are not affected by either changing L or using other algorithms [21] to perform the numerical simulations. 19