Fractal Growth with Quenched Disorder

Abstract

In this lecture we present an overview of the physics of irreversible fractal growth process, with particular emphasis on a class of models characterized by {\em quenched disorder}. These models exhibit self-organization, with critical properties developing spontaneously, without the fine tuning of external parameters. This situation is different from the usual critical phenomena, and requires the introduction of new theoretical methods. Our approach to these problems is based on two concepts, the Fixed Scale Transformation, and the quenched-stochastic transformation, or Run Time Statistics (RTS), which maps a dynamics with quenched disorder into a stochastic process. These methods, combined together, allow us to understand the self-organized nature of models with quenched disorder and to compute analytically their critical exponents. In addition, it is also possible characterize mathematically the origin of the dynamics by {\em avalanches} and compare it with the {\em continuous growth} of other fractal models. A specific application to Invasion Percolation will be discussed. Some possible relations to glasses will also be mentioned.