Cluster-size distribution of self-affine fractals

The relationship between structural properties of growing patterns and their cluster statistics is discussed. We consider the cases in which an entire pattern grown from a $d_s$-dimensional substrate is self-similar with a fractal dimension D, whereas individual clusters constituting it are self-affine with exponents ${\nu }_{?}$ and ${\nu }_{\perp }$ for the scaling behavior of the height and width, respectively. Based on a scaling assumption for the cluster-size distribution function ($n_s$∼$s^{\mathrm{-}\tau }$; s is the cluster size), the scaling relation τ=2-${\nu }_{?}$(D-$d_s$) is derived. It is found that the relation is in excellent agreement with almost all available simulation data for growth models investigated so far.