Percolation thresholds and fractal dimensions for square and cubic lattices with long-range correlated defects

TL;DR

This study investigates how long-range correlated defects affect percolation thresholds and fractal dimensions in square and cubic lattices, revealing that thresholds are system-dependent while fractal dimensions are universal and influenced by correlation strength.

Contribution

The paper provides high-precision measurements of percolation thresholds and fractal dimensions under long-range correlations, highlighting the universality of fractal dimensions regardless of correlation details.

Findings

Percolation thresholds are sensitive to correlation metrics.

Fractal dimension $d_f$ is universal and unaffected by correlation metrics.

In 2D, $d_f$ increases with correlation strength, approaching 2.

Abstract

We study long-range power-law correlated disorder on square and cubic lattices. In particular, we present high-precision results for the percolation thresholds and the fractal dimension of the largest clusters as function of the correlation strength. The correlations are generated using a discrete version of the Fourier filtering method. We consider two different metrics to set the length scales over which the correlations decay, showing that the percolation thresholds are highly sensitive to such system details. By contrast, we verify that the fractal dimension $d_{\rm f}$ is a universal quantity and unaffected by the choice of metric. We also show that for weak correlations, its value coincides with that for the uncorrelated system. In two dimensions we observe a clear increase of the fractal dimension with increasing correlation strength, approaching $d_{\rm f}\rightarrow 2$. The onset of this change does not seem to be determined by the extended Harris criterion.