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## V. DIMENSIONAL ANALYSIS AND NONLINEAR DYNAMICS 

Our results so far allow us to develop a consistent dimensional analysis of radial equations. We start with Eq. (4) performing the scale transformation ρ → b $^{α}$ρ and t → b $^{z}$t . The angular variables are dimensionless and this, as specified by the factor J , fixes the exponent z = 1 /γ . The stochastic term becomes scale invariant for α = 1 / (2 γ ) − d/ 2 and the diffusion irrelevant for γ > 1 /ζ : this agrees with our previous analysis using the explicit solutions of the linear equations. We can now use this technique to get some insight on nonlinear equations. The Kardar-Parisi-Zhang (KPZ) equation [23] for the radial surface fluctuation can be written as 

<img src='content_image/357058.jpg'>

We note that scaling arguments were successfully employed in determining the critical be- havior of KPZ with a time dependent coefficient of the nonlinearity [24]. If we select the same values of α and z as in the linear case in order to make the stochastic term scale invariant we find that diffusion and nonlinearity are irrelevant for γ > 1 / 2 and γ > 3 / (4+ d ) respectively. This agrees with the na¨ıf dimensional analysis of KPZ and thus we cannot expect it to be correct. In fact, the na¨ıf dynamical exponent z = (4 + d ) / 3 found by means of a dimensional analysis of the KPZ nonlinearity does not agree with the one measured in simulations and successfully predicted by the dynamic renormalization group analysis in one dimension [7]. It is plausible that the nonlinearity interacts with the noise to yield a correlation different from (6) (or from (11) if dilution is included). In order to guess what form the correlation would have we can try to get some illumination from scaling (2). Taking the short time limit in (2) we find 

<img src='content_image/357056.jpg'>

where the temporal prefactor expresses the time dependence of the variance of the interface center of mass position [16]. Using this form we modify, for instance, Eq. (11) to the new correlation 

<img src='content_image/357055.jpg'>

which yields the exponents α = (2 β$_{c}$ + d/z$_{c}$ ) / (2 γ ) − d/ 2, z = 1 /γ and β$_{r}$ = (2 β$_{c}$ + d/z$_{c}$ ) / 2 − dγ/ 2. In this case we have the diffusion is irrelevant whenever γ > 1 / 2 and the nonlinearity is irrelevant if γ > 1 /z$_{c}$ , where we have employed the relation α$_{c}$ + z$_{c}$ = 2 in the derivation of 

