Tip Splittings and Phase Transitions in the Dielectric Breakdown Model: Mapping to the DLA Model

Abstract

We show that the fractal growth described by the dielectric breakdown model exhibits a phase transition in the multifractal spectrum of the growth measure. The transition takes place because the tip-splitting of branches forms a fixed angle. This angle is eta dependent but it can be rescaled onto an ``effectively'' universal angle of the DLA branching process. We derive an analytic rescaling relation which is in agreement with numerical simulations. The dimension of the clusters decreases linearly with the angle and the growth becomes non-fractal at an angle close to 74 degrees (which corresponds to eta= 4.0 +- 0.3).