FIXED SCALE TRANSFORMATION APPROACH TO CRITICAL FLUCTUATIONS AND FRACTAL GROWTH

A geometrical approach to the study of critical phenomena calls for a fractal description of critical fluctuations. The fixed scale transformation (FST) approach introduces such a description, incorporating both the scale and translational invariance properties of the critical state. Besides providing a theoretical understanding of the genesis of fractal structures, it enables one to compute the fractal dimension of the critical clusters to a high degree of accuracy, in terms of the critical statistics of the underlying model. Laplacian models of irreversible fractal growth, to which the FST approach was first applied, are presented within the general framework of critical phenomena. This geometrical approach links the growth rules, which are fully nonlocal in both space and time (measured in number of particles added), to the morphology of the grown cluster and its scaling behaviour. Possible directions towards a more comprehensive theory of fractal growth are discussed.