content_image/335579.jpg content_image/335577.jpg content_image/335576.jpg content_image/335575.jpg
finds [16] 

<img src='content_image/335579.jpg'>

This correlation is invariant to the transformation ℓ → bℓ , t → b $^{z}$t and ρ → b $^{α}$ρ for z = 1 /γ =: z$_{r}$ and α = 1 / (2 γ ) − d/ 2 =: α$_{r}$ ( α = 0 =: α$_{r}$ ) when γd < 1 ( γd ≥ 1). We define β$_{r}$ := α/z = α$_{r}$/z$_{r}$ = (1 − γd ) / 2 ( β$_{r}$ = 0) when γd < 1 ( γd ≥ 1). First, the functional form of this correlation is not reducible to scaling form (2), and second, the exponents are totally different (this makes precise the analysis carried out in [13]). In summary, long range radial surface fluctuations are not describable in terms of scaling (2). 

## III. DILUTION AND UNIVERSALITY CLASS BIFURCATION 

It is clarifying to consider a simplified problem: an abstraction in which Eq. (3) is placed on a growing domain [20]. If this is realized by means of the transformation x → ( t/t$_{0}$ ) $^{γ}$x one finds ζγ dγ/ 2 

<img src='content_image/335577.jpg'>

where the mass source is explicitly included and we assume fast growth γ > 1 /ζ and the rough interface inequality ζ > d . In this case it can be shown that the correlation obeys scaling (2) for short length scales | x − x $^{′}$|  ≪ t (1 − ζγ ) /ζ and for long length scales | x − x $^{′}$|  ≫ t (1 − ζγ ) /ζ it is [20] 

<img src='content_image/335576.jpg'>

where y = ( t/t$_{0}$ ) $^{γ}$x is the counterpart of the arc-length coordinate. This correlation is scale invariant for the exponents { α$_{r}$, z$_{r}$ } but not for the exponents { α$_{c}$, z$_{c}$ } . 

This way of realizing domain growth agrees with reparametrization invariance [16], but is not unique. It makes space and mass grow simultaneously [16], while it is possible to realize domain growth at constant mass by introducing a dilution term [20, 21] 

<img src='content_image/335575.jpg'>

