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 f ield theory and renormalization group do not seem to be suitable. The fixed-scale transformation (FST) is a theoretical scheme of a novel ty pe that can deal with such problems in a reasonably systematic way. Th e main idea is to focus on the irreversible dynamics at a given scale and to compute accurately the nearest-neighbor correlations at this sc ale by suitable lattice path integrals. The next basic step is to iden tify the scale-invariant dynamics that refers to coarse-grained variab les of arbitrary scale. The use of scale-invariant growth rules allows us to generalize these correlations to coarse-grained cells of any si ze and therefore to compute the fractal dimension. The basic point is to split the long-time limit (t-->infinity) for the dynamical process at a given scale that produces the asymptotically frozen structure, fr om the large-scale limit (r-->infinity) which defines the scale-invari ant dynamics. In addition, by working at a fixed scale with respect to dynamical evolution, it is possible to include the fluctuations of bo undary conditions and to reach;a remarkable level of accuracy for a re al-space method. This new framework is able to explain the self-organi zed critical nature and the origin of fractal structures in irreversib le-fractal-growth models, it also provides a rather systematic procedu re for the analytical calculation of the fractal dimension and other c ritical exponents. The FST method can be naturally extended to a varie ty of equilibrium and nonequilibrium models that generate fractal stru ctures.