content_image/346659.jpg content_image/346656.jpg content_image/346654.jpg content_image/346655.jpg
In this case the correlation is 

<img src='content_image/346659.jpg'>

which is scale invariant for both { α$_{c}$, z$_{c}$ } and { ˜ α$_{r}$ = 1 / (2 γ ) − d/ 2 , z$_{r}$ } . The first invariance can be used to argue that this correlation is a direct consequence of scaling (2), but the scaling form (10) is definitely ambiguous in view of the second invariance. Both sort of dynamics, with and without dilution, are physically realizable; while dilution dynamics is apparently better suited to describe some natural processes, no-dilution dynamics seems to agree better with the numerical simulation of some discrete models as discussed in [16]. Together with the difference in the roughness exponent, very pronounced differences in the auto-correlation and persistence exponents separate the interface behavior in presence and absence of dilution [20]. This reveals that the non-uniqueness of the extension of a Langevin equation to a growing domain implies in turn a universality class bifurcation. 

## IV. GEOMETRIC CONSTRAINTS 

One can apply what has been learned in the previous section about growing domains to the radial situation. Including dilution in the radial equation for interface motion yields a different dynamics from the one established by reparametrization invariance. The correlation in this case becomes [16] 

<img src='content_image/346656.jpg'>

which is scale invariant for the exponents { ˜ α$_{r}$, z$_{r}$ } but not for { α$_{c}$, z$_{c}$ } . The difference among α$_{r}$ and ˜ α$_{r}$ originates in the memory effects that affect the surface described by reparametrization invariance and disappear by virtue of dilution [16, 20, 22]. The difference between the sets { α$_{r}$, z$_{r}$ } and { ˜ α$_{r}$, z$_{r}$ } , and the classical set { α$_{c}$, z$_{c}$ } is of a different nature. Note that the Langevin dynamics on a growing domain and influenced by dilution gave rise to the random deposition correlation (10): 

This correlation is invariant to the transformation 

<img src='content_image/346654.jpg'>

<img src='content_image/346655.jpg'>

