Fractal dimension of intersection sets in the dielectric breakdown model

Abstract

Clusters grown with the dielectric breakdown model (DBM) in the cylinder geometry show two growth phases: a scaling regime for cluster heights smaller than the cylinder circumference and a subsequent steady state, which is translational invariant in the main growth direction. The box-counting dimension of one-dimensional intersection sets of the clusters is studied for six different values of the growth parameter eta and four cylinder circumferences. The author finds that in the scaling regime this dimension depends on the height at which the intersection is made and the clusters are thus inhomogeneous. The results also show that clusters in the steady state are translational invariants in the main growth direction and self-similar in the direction perpendicular to it. A comparison between his findings and the theoretical results obtained from the fixed scale transformation approach by Pietronero et al. (1988), shows good agreement.