Fixed scale transformation applied to diffusion limited aggregation and dielectric breakdown model in three dimensions

We extend the method of the fixed scale transformation (FST) to the case of fractal growth in three dimensions and apply it to diffusion limited aggregation and to the dielectric breakdown model for different values of the parameter η. The scheme is formally similar to the two-dimensional case with the following technical complications: (i) The basis configurations for the fine graining process are five (instead of two) and consist of 2 × 2 cells. (ii) The treatment of the fluctuations of boundary conditions is far more complex and requires new schemes of approximations. In order to test the convergency of the theoretical results we consider three different schemes of increasing complexity. For DBM in three dimensions the computed values of the fractal dimension for η = 1, 2 and 3 result to be in very good agreement with corresponding values obtained by computer simulations. These results provide an important test for the FST method as a new theoretical tool to study irreversible fractal growth.