Theory of Laplacian fractals: Diffusion limited aggregation and dielectric breakdown model

We describe a new theoretical approach that clarifies the origin of fractal structures in irreversible growth models based on the Laplace equation and a stochastic field. This new theory provides a systematic method for the calculation of the fractal dimension D and of the multifractal spectrum of the growth probability (ƒ(α)). A detailed application to the dielectric breakdown model and diffusion limited aggregation in two dimensions is presented. Our approach exploits the scale invariance of the Laplace equation that implies that the structure is self-similar both under growth and scale transformation. This allows one to introduce a Fixed Scale Transformation (instead of coarse graining as in the renormalization group theory) that defines a functional equation for the fixed point of the distribution of basic diagrams used in the coarse graining process. For the calculation of the matrix elements of this transformation one has to consider an infinite, but rapidly convergent, number of processes that occurs outside a considered diagram.