The fixed-scale transformation approach to fractal growth

Irreversible fractal-growth models like diffusion-limited aggregation (DLA) and the dielectric breakdown model (DBM) have confronted us with theoretical problems of a new type for which standard concepts like field theory and renormalization group do not seem to be suitable. The fixed-scale transformation (FST) is a theoretical scheme of a novel type that can deal with such problems in a reasonably systematic way. The main idea is to focus on the irreversible dynamics at a given scale and to compute accurately the nearest-neighbor correlations at this scale by suitable lattice path integrals. The next basic step is to identify the scale-invariant dynamics that refers to coarse-grained variables of arbitrary scale. The use of scale-invariant growth rules allows us to generalize these correlations to coarse-grained cells of any size and therefore to compute the fractal dimension. The basic point is to split the long-time limit ($t\to \infty$) for the dynamical process at a given scale that produces the asymptotically frozen structure, from the large-scale limit ($r\to \infty$) which defines the scale-invariant dynamics. In addition, by working at a fixed scale with respect to dynamical evolution, it is possible to include the fluctuations of boundary conditions and to reach a remarkable level of accuracy for a real-space method. This new framework is able to explain the self-organized critical nature and the origin of fractal structures in irreversible-fractal-growth models. It also provides a rather systematic procedure for the analytical calculation of the fractal dimension and other critical exponents. The FST method can be naturally extended to a variety of equilibrium and nonequilibrium models that generate fractal structures.