Model for Anisotropic Directed Percolation

TL;DR

This paper introduces a simulation model for anisotropic directed percolation in 2D media, revealing invariants in bond number and fractal dimension that extend isotropic percolation results to anisotropic cases.

Contribution

The study demonstrates that Sinai's theorem and related invariants hold for anisotropic directed percolation, extending fundamental percolation theory.

Findings

Average bonds per site is invariant at 2.8 for all anisotropy levels.

Fractal dimension remains approximately 1.71 regardless of anisotropy.

Invariant properties suggest universality in anisotropic directed percolation.

Abstract

We propose a simulation model to study the properties of directed percolation in two-dimensional (2D) anisotropic random media. The degree of anisotropy in the model is given by the ratio $\mu$ between the axes of a semi-ellipse enclosing the bonds that promote percolation in one direction. At percolation, this simple model shows that the average number of bonds per site in 2D is an invariant equal to 2.8 independently of $\mu$. This result suggests that Sinai's theorem proposed originally for isotropic percolation is also valid for anisotropic directed percolation problems. The new invariant also yields a constant fractal dimension $D_{f} \sim 1.71$ for all $\mu$, which is the same value found in isotropic directed percolation (i.e., $\mu = 1$).