Scale Invariance in Percolation Theory and Fractals

TL;DR

This paper explores the fractal and scale-invariant properties of percolation in the complex plane, revealing connections to Mandelbrot transformations and fixed points in impedance mappings.

Contribution

It demonstrates that percolation probability exhibits fractal behavior and connects percolation theory to complex dynamics and Mandelbrot transformations.

Findings

Percolation probability shows fractal distribution in the complex plane.

The percolation problem reduces to Mandelbrot transformation.

Fixed points emerge in impedance mappings with fractal step distributions.

Abstract

The properties of the similarity transformation in percolation theory in the complex plane of the percolation probability are studied. It is shown that the percolation problem on a two-dimensional square lattice reduces to the Mandelbrot transformation, leading to a fractal behavior of the percolation probability in the complex plane. The hierarchical chains of impedances, reducing to a nonlinear mapping of the impedance space onto itself, are studied. An infinite continuation of the procedure leads to a fixed point. It is shown that the number of steps required to reach a neighborhood of this point has a fractal distribution.