convergence. This is so at least in situations in which the height-difference correlation is well defined; otherwise a different scaling is possible even locally in space [16]. This makes precise and generalizes our previous results [13] in the following way: a fast growth rate will make the interface weakly converge to a hypersphere. We have further found the scaling relation α = β$_{r}$z , which quantitatively relates the center of mass fluctuations, growth rate and rate of (weak) convergence to/divergence from the hyperspherical profile. These results, together with our previous ones [16, 20], suggest the following generalization of the critical exponents for arbitrary values of γ which should be valid for linear models as well as some nonlinear equations as the VLDS one, which present a conserved drift and its critical behavior is accessible to dimensional analysis. For the KPZ equation, which nonlinearity introduces both mass and fluctuations into the system, we expect a different behavior. As we have seen, the amount of mass entering the interface (and of course the way it is introduced) may change the critical behavior. We have explicitly shown this by means of the linear models in absence and presence of dilution, which gave respectively rise to correlations (6) and (11). We have summarized the different morphologies that can appear in Table I (see also Fig. 2 in [16]). In the first column, for small γ , we have placed the regime in which the classical values of the exponents are recovered. The other three columns represent situations in which this is not the case. The “roughness” column describes the situation in which the radial fluctuations do not average each other out for long times, so in this regime the overall appearance of the cluster deviates from the hyperspherical profile. The “super-roughness” column states a particular case of this last regime, in which the amplitude of the radial fluctuations grows faster than the arc-length sustaining them. In both cases roughness should be understood as the degree of weak divergence (as opposed to weak convergence) from the hyperspherical profile. Finally, the last column describes the regime in which the weak convergence to the hyperspherical profile is reached asymptotically in time. 12 γ < 1 z$_{c}$ 1 z$_{c}$ < γ < 1 d 1 z$_{c}$ < γ < 1 d +2 γ > max { 1 $_{d}$, 1 z$_{c}$ } Classical exponents Roughness Super-roughness Weak convergence to the hyperspherical profile TABLE I: Morphologies for different spatial dimensions and exponents. α = z − d 2 , β := α z = 1 2 − d 2 z , for z = min$^{{}$ z$_{c}$, 1 γ$^{}}$ , (19)