finds [16] 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). 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 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] 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] 7 ∂$_{t}$h = − D $^{(}$t$_{0}$ t ) ζγ |∇| $^{ζ}$h − dγ t h + γF t γ − 1 + $^{(}$t$_{0}$ t ) dγ/ 2 ξ. (9) C ( y, y $^{′}$, t ) ∼      tδ ( y − y $^{′}$) if γd < 1 , t ln( t ) δ ( y − y $^{′}$) if γd = 1 , t $^{γd}$δ ( y − y $^{′}$) if γd > 1 , (8) ∂$_{t}$h = − D $^{(}$t$_{0}$ t ) ζγ |∇| $^{ζ}$h + γF t γ − 1 + $^{(}$t$_{0}$ t ) dγ/ 2 ξ, (7) III. DILUTION AND UNIVERSALITY CLASS BIFURCATION C ( ℓ, ℓ $^{′}$, t ) ∼      tδ ( ℓ − ℓ $^{′}$) /J ( t − $^{γ}$ℓ ) if γd < 1 , t ln( t ) δ ( ℓ − ℓ $^{′}$) /J ( t − $^{γ}$ℓ ) if γd = 1 , t $^{γd}$δ ( ℓ − ℓ $^{′}$) /J ( t − $^{γ}$ℓ ) if γd > 1 . (6)