Scaling properties for the growth probability measure and harmonic measure of fractal structures

Computer simulations have been used to explore the scaling properties of the growth probability measures and harmonic measure for a variety of fractal objects. In all cases we have studied the normalized distribution of growth or contact probabilities, N(P), for clusters with different masses, M, can be scaled onto a single curve using the scaling form ln(PN(P)lnM]=ln(M)g‘(ln(P)/ln(M)). The scaling function g’(x) is related to the function f(α) of Halsey et al. by g’(x)=$D^{\mathrm{-}1}$f(-Dx), where D is the fractal dimensionality of the set on which the measure resides. Here f(α) is the fractal dimensionality of the subset which supports singularities of type α. Similarly, for diffusion-limited aggregation on strips of width L and columns of area L×L we find that ln[PN(P)=ln(L)h(ln(P)/ln(L)), where h(x)=f(-x) and N(P)δP is the number of sites with growth probabilities (also harmonic measure probabilities in this case) in the range P to P+δP. Our results indicate that the ideas recently developed by Halsey et al. and the earlier ideas of Mandelbrot are applicable to a broad range of processes on fractal aggregates as well as to dynamic systems with fractal properties.