Percolation fractal exponents without fractal geometry

TL;DR

This paper demonstrates that certain percolation power laws, including the 7/4 exponent, remain valid even when the cluster frontiers are no longer fractal, indicating a fundamental scaling law independent of fractal geometry.

Contribution

It reveals that gradient percolation power laws persist beyond fractal regimes, challenging the traditional link between fractal properties and critical exponents.

Findings

The 7/4 exponent applies to gradient percolation frontiers regardless of gradient magnitude.

A single power law describes the entire size range of the gradient percolation frontier.

The power law becomes a scaling law in the large system limit.

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 keeps its value in describing the width and length of gradient percolation frontiers whatever the gradient value. Its origin is then not to be found in the thermodynamic limit. 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 from size 1 to infinity which describes the gradient percolation frontier, this law becoming a scaling law in the large system limit.