Statistics of the critical percolation backbone with spatial long-range correlations

TL;DR

This paper investigates how spatial long-range correlations affect the statistics of the backbone in two-dimensional percolation networks, revealing a universal scaling form and linking the backbone exponent to fractal dimension.

Contribution

The study introduces a scaling ansatz for backbone mass distribution in correlated percolation and demonstrates its applicability through extensive simulations, connecting the backbone exponent to fractal properties.

Findings

The backbone mass distribution follows a specific scaling form with a cutoff function.

The scaling form applies to both correlated and uncorrelated networks.

The backbone exponent is related to the fractal dimension and depends on correlation strength.

Abstract

We study the statistics of the backbone cluster between two sites separated by distance $r$ in two-dimensional percolation networks subjected to spatial long-range correlations. We find that the distribution of backbone mass follows the scaling {\it ansatz}, $P(M_B)\sim M_B^{-(\alpha+1)}f(M_B/M_0)$, where $f(x)=(\alpha+ \eta x^{\eta}) \exp(-x^{\eta})$ is a cutoff function, and $M_0$ and $\eta$ are cutoff parameters. Our results from extensive computational simulations indicate that this scaling form is applicable to both correlated and uncorrelated cases. We show that the exponent $\alpha$ can be directly related to the fractal dimension of the backbone $d_B$, and should therefore depend on the imposed degree of long-range correlations.