Fractal Growth with Quenched Disorder

TL;DR

This paper reviews the physics of irreversible fractal growth with quenched disorder, introducing new theoretical methods to analyze self-organization, critical properties, and avalanches, with applications to invasion percolation.

Contribution

It presents novel analytical methods, Fixed Scale Transformation and RTS, to understand and compute critical exponents in models with quenched disorder, highlighting their self-organized criticality.

Findings

Analytical computation of critical exponents for models with quenched disorder.

Characterization of avalanche dynamics in fractal growth.

Application of methods to invasion percolation and potential links to glasses.

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.