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 Mˣ and whose size (number of sites) grows as Mᵍ⁽ˣ⁾. 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^-1f(-Dx).