Fixed scale transformation approach to the multifractcal properties of the growth probabilities in the dielectric breakdown model

Abstract

The separation of the properties of the growth probability distribution in two different contributions, as discussed in the previous paper, corresponds naturally to the approximation scheme of the fixed scale transformation (FST) method. The growth probabilities used to compute the FST matrix elements represent the essential elements of the multiplicative process that gives rise to the regular part (the only one relevant to the growth process) of the multifractal spectrum. The FST uses these probabilities directly without the need of introducing a multifractal spectrum explicitly. This, however, can be obtained as a by-product of the FST method. We present here analytical calculations for the regular part of the multifractal spectrum of the dielectric breakdown model with different values of the parameter ν. The results are good for η ⩾ 1 and less accurate for η < 1. In fact for small η values, and in order to recover the Eden limit, it is necessary to go to higher order and possibly to include self-affine properties explicitly.