Properties of the growth probability for the dielectric breakdown model in cylinder geometry

We study the properties of the growth probabilities for diffusion limited aggregation and the dielectric breakdown model in the steady state regime of the cylinder geometry. The results show a rather unambiguous picture with the following properties: The projection of the growth probability along the growth direction is exponential, contrary to the Gaussian behavior of the radial case. One can distinguish two regions, one with simple multifractal properties corresponding to the growing zone and a second one which accounts for the exponential decay of the growth probability. This situation can be explained in terms of a first order transition in the multifractal spectrum at q = 1. This corresponds to a different picture with respect to those that have been proposed in the literature. The properties of the growing interface could be universal while the small probability part is non-universal and it depends on the particular geometry. However, we can show that this part is irrelevant with respect to the growth process, even though it is determined by it.