Scaling of the growth probability measure for fractal structures

The growth probability measure has been determined for a family of screened growth models with a continuously tunable fractal dimensionality. The distribution of growth probabilities N(P) for clusters of different masses M can be scaled onto a single curve g(x) using the scaling form ln[PN(P)lnM]=ln(M)g(ln(P)/ln (M)). Each point in the scaling function g(x) corresponds to a part of the growth probability measure whose probability P grows as $M^x$ and whose size (number of sites) grows as $M^{g\left(x\right)}$. The function g(x) is related to the function f(α) of Halsey et al. which associates a fractal dimensionality f(α) with that part of the measure which consists of singularities of strength α by g(x)=$D^{\mathrm{-}1}$f(-Dx).