Self-Organized Percolation Power Laws with and without Fractal Geometry in the Etching of Random Solids

TL;DR

This paper demonstrates that percolation power laws and fractal properties are conserved across a wide range of gradient values, including non-fractal frontiers, with implications for understanding etching processes and underlying conservation laws.

Contribution

It reveals that gradient percolation power laws hold even for non-fractal frontiers and extends this understanding to etching models, suggesting a universal underlying conservation law.

Findings

Power laws are conserved for all gradient values.

The fractal dimension controls system geometry regardless of gradient.

Numerical and analytical results indicate a universal power law.

Abstract

Classically, percolation critical exponents are linked to the power laws that characterize percolation cluster fractal properties. It is found here that the gradient percolation power laws are conserved even for extreme gradient values for which the frontier of the infinite cluster is no more fractal. In particular the exponent 7/4 which was recently demonstrated to be the exact value for the dimension of the so-called "hull" or external perimeter of the incipient percolation cluster, controls the width and length of gradient percolation frontiers whatever the gradient magnitude. This behavior is extended to previous model studies of etching by a finite volume of etching solution in contact with a disordered solid. In such a model, the dynamics stop spontaneously on an equilibrium self-similar surface similar to the fractal frontier of gradient percolation. It is shown that the power laws describing the system geometry involves also the fractal dimension of the percolation hull, whatever the value of the dynamically generated gradient, i.e. even for a non-fractal frontier. The comparison between numerical results and the exact results that can be obtained analytically for extreme values of the gradient suggests that there exist a unique power law valid from the smallest possible scale up to infinity. These results suggest the possible existence of an underlying conservation law, relating the length and the statistical width of percolation gradient frontiers.