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

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 a 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 $d_{\mathrm{f}}$ 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 $d_{\mathrm{f}}\to 2$. The onset of this change does not seem to be determined by the extended Harris criterion.

Figure 1

Illustration of long-range correlated defects on a ${2048}^2$ lattice with correlation parameter $a=0.5$. (a) Continuous correlated Gaussian random variables. The color reflects the value at the respective lattice site. (b) Corresponding lattice of discrete variables at the finite-size percolation threshold $p_c^L=0.522$ with defects shown in black.

Figure 2

Correlation function $C\left(\mathbf{r}\right)$ compared to the measured site-site correlation function ${〈C_{\mathbf{r}}〉}_R$ of continuous variables along the $x$ direction on a ${16}^2$ lattice with $R={10}^6$ disorder replicas. The continuous random variables are obtained via a discrete Fourier transform of $C_{\mathbf{r}}$ and satisfy ${\sigma }_{{\tau }_{}}^2={〈C_0〉}_R=1$.

Figure 3

Normalized correlation function of discrete random variables with long-range power-law correlation at the percolation threshold $p_c\left(a\right)$ (see Table 2) on a ${1024}^2$ lattice averaged over ${10}^4$ disorder replicas. Continuous correlated random variables were obtained as described in Sec. 2. The mapping to discrete variables is performed via a global approach (open symbols), i.e., on the level of the disorder average [see Eq. (14)], and via a local approach, i.e., on the level of each disorder realization (solid symbols). The measured site-site correlation function ${〈C_{\mathbf{r}}〉}_R$ along the $x$ axis is normalized with respect to the variance ${〈C_0〉}_R$ for discrete site variables.

Figure 4

(a) Measured percolation thresholds $p_c^L$ for varying lattice size $L$ and different values of the correlation parameter $a$ in two dimensions (Euclidean metric). Lines show the best fits of Eq. (16) to the data, whose intercepts represent our estimates for the infinite-system value $p_c$. (b) Estimates for $p_c$ as function of the correlation strength for the square lattice with distances measured in the Euclidean metric (squares) and the Manhattan metric (crosses). The horizontal line shows the value for the system without correlations.

Figure 5

(a) Measured percolation thresholds $p_c^L$ for varying lattice size $L$ and different values of the correlation parameter $a$ in three dimensions (Euclidean metric). Lines show the best fits of Eq. (18) to the data, whose intercepts represent our estimates for the infinite-system value $p_c$. (b) Estimates for $p_c$ as a function of the correlation strength for the cubic lattice with distances measured in the Euclidean metric (squares) and the Manhattan metric (crosses). The horizontal line shows the value for the system without correlations.

Figure 6

(a) Average volume fraction of the largest cluster vs lattice extension in two dimensions plotted for different correlations $a$ on a double-logarithmic scale (Euclidean metric). Colored lines are least-squares fits to Eq. (19). The black line represents the behavior of the uncorrelated system with a slope of $91/48-2$. (b) Overview of our estimates for the fractal dimensions as function of $a$ with the horizontal line again corresponding to the uncorrelated system.

Figure 7

(a) Average volume fraction of the largest cluster vs lattice extension in three dimensions plotted for the exemplary case $a=2$ on a double-logarithmic scale. Colored lines are the simultaneous fits according to Eq. (20); the black line represents the behavior of the uncorrelated system with a slope of $2.52295-3$. (b) Overview of our estimates for the fractal dimensions as function of $a$ with the horizontal line again corresponding to the uncorrelated system. Our fitting did not work properly for very strong correlations ($a\le 1$).

Figure 8

(a) Demonstration of the effect of the zero cutoff $S_{\mathbf{k}}\to {\tilde{S}}_{\mathbf{k}}$ on the measured correlation function ${〈C_{\mathbf{r}}〉}_R$ in three dimensions for $a=0.5$. Also shown is the prediction ${\tilde{C}}_{\mathbf{r}}$ for the asymptotic limit from Eq. (B2). (b) Plot of the deviations from the desired function $C\left(\mathbf{r}\right)$.