Percolation with long-range correlated disorder

Long-range power-law correlated percolation is investigated using Monte Carlo simulations. We obtain several static and dynamic critical exponents as functions of the Hurst exponent $H$, which characterizes the degree of spatial correlation among the occupation of sites. In particular, we study the fractal dimension of the largest cluster and the scaling behavior of the second moment of the cluster size distribution, as well as the complete and accessible perimeters of the largest cluster. Concerning the inner structure and transport properties of the largest cluster, we analyze its shortest path, backbone, red sites, and conductivity. Finally, bridge site growth is also considered. We propose expressions for the functional dependence of the critical exponents on $H$.

Figure 1

Triangular lattice stripe of size $L=4$ and aspect ratio $A=2$.

Figure 2

Convergence of the percolation threshold estimator $p_{c,J}$. The difference between the estimator and the threshold $|p_{c,J}-1/2|$ is shown as a function of the lattice size $L$ for $H=-1$, $-0.7$, $-0.4$, and $-0.1$. The data are shifted vertically to improve visibility. Results are averages over ${10}^5$ samples. We keep track of the cluster properties with the labeling method proposed by Newman and Ziff [39, 40], as in Ref. [65].

Figure 3

(a) Fraction of sites in the largest cluster $s_{\max }/N$ as function of the lattice size $L$ for different values of $H$. (b) Second moment of the cluster size distribution $M_2^{\prime }$ as function of $L$ for the same values of $H$ as in (a). The data is shifted vertically to improve visibility. Solid black lines are guides to the eye. Results are averages over at least ${10}^4$ samples.

Figure 4

(a) Fractal dimension of the largest cluster $d_f$ and critical exponent ratio ${\gamma }_H/{\nu }_H$ as a function of the Hurst exponent $H$, where ${\gamma }_H$ is the susceptibility critical exponent and ${\nu }_H$ is the correlation-length critical exponent. For $H>-1/3$, the solid lines show the expressions of Eqs. (8) and (14). (b) With $d_f=d-{\beta }_H/{\nu }_H$, where ${\beta }_H$ is the order parameter critical exponent, the hyperscaling relation reads $2=d={\gamma }_H/{\nu }_H+2{\beta }_H/{\nu }_H$ [1]. One observes that the data agree, within error bars, with the hyperscaling relation.

Figure 5

Complete and accessible perimeter. The blue (filled) sites of the triangular lattice are part of the largest cluster, while the white (empty) sites are unoccupied. Bonds of the dual lattice are shown as dashed lines. Assume that the largest cluster percolates in the vertical direction and does not touch the left or right boundaries of the lattice. Consider a walker starting on the bottom left side of the lattice, which never visits a bond twice and traces the complete perimeter, turning left or right depending on which of the two available bonds separates an occupied from an empty site. The complete perimeter is fully determined when the top side of the lattice is reached. Performing the same walk but with the additional constraint that fjords with diameter $\le \sqrt{3}/3$ (in lattice units) are not accessible yields the accessible perimeter. The solid green (thick) lines on the honeycomb lattice form the accessible perimeter, while dashed green (thick) lines indicate bonds that are part of the complete perimeter but not of the accessible one. A similar walk yields the two perimeters on the right-hand side of the cluster.

Figure 6

Snapshots of typical complete and accessible perimeters. The accessible perimeter is shown in bold solid blue lines. In addition, the parts of the complete perimeter that do not belong to the accessible perimeter are drawn with thin black lines. The snapshots are taken for (a) $H=-1$, (b) $-0.5$, (c) $-0.25$, and (d) 0, on a lattice of (vertical) length $L=128$.

Figure 7

Fractal dimension of the complete perimeter $d_{\mathrm{CP}}$ and of the accessible perimeter $d_{\mathrm{AP}}$ as a function of the Hurst exponent $H$. For $H=-1$ (uncorrelated), our results $d_{\mathrm{CP}}=1.75\pm 0.02$ and $d_{\mathrm{AP}}=1.34\pm 0.02$ are in agreement with values previously reported [53, 68, 69, 72, 78, 79, 80]. With increasing $H$, both fractal dimensions seem to approach $3/2$, which is compatible with the data of Kalda et al. [36, 76, 77, 81]. In the range $-1/{\nu }_{2\mathrm{D}}^{\mathrm{uncorr}}\le H\le 0$, the solid lines show the expressions $d_{\mathrm{CP}}=3/2-H/3$ and $d_{\mathrm{AP}}=(9-4H)/(6-4H)$. The insets show the length of the complete and of the accessible perimeter as a function of the lattice size $L$ for the values of $H$ shown in the main plot.

Figure 8

Yardstick method to measure the fractal dimension of the complete perimeter. The number of sticks needed to follow the perimeter $S$ is shown as a function of the stick length $m$, for different lattice sizes $L$, and $H=0$. The numerical value of the complete perimeter fractal dimension $d_{\mathrm{CP}}\left(H\right)$ obtained with the yardstick method, $d_{\mathrm{CP}}\left(0\right)=1.49\pm 0.03$, agrees, within error bars, with the results of the analysis of the local slopes of the perimeter length (see Fig. 7), as well as with the literature [36, 76, 77].

Figure 9

Left hand side of the duality relation for cluster perimeters, $(d_{\mathrm{AP}}-1)(d_{\mathrm{CP}}-1)=1/4$ [80, 85], as function of the Hurst exponent $H$.

Figure 10

Fractal dimension of the shortest path $d_{\mathrm{SP}}$ of the largest cluster as a function of the Hurst exponent $H$. The inset shows the number of sites in the shortest path as a function of the lattice size $L$ for the same value of $H$ as in the main plot. For uncorrelated disorder, i.e., $H=-1$, we find $d_{\mathrm{SP}}=1.130\pm 0.005$, in agreement with the literature [87, 88, 89, 90]. With increasing Hurst exponent, $d_{\mathrm{SP}}$ approaches unity [91]. This behavior is due to the backbone becoming increasingly compact as $H$ approaches 0 (see Fig. 11). The solid line is the graph of the proposed behavior of the shortest path fractal dimension $d_{\mathrm{SP}}\left(H\right)=147/130-(3/4+H)/(195/34+H)$, for $-3/4\le H\le 0$, and $d_{\mathrm{SP}}(-1\le H\le -1/{\nu }_{2\mathrm{D}}^{\mathrm{uncorr}})=d_{\mathrm{SP}}(-1/{\nu }_{2\mathrm{D}}^{\mathrm{uncorr}})$.

Figure 11

Fractal dimension of the backbone $d_{\mathrm{bb}}$ as a function of the Hurst exponent $H$. With increasing $H$, the backbone becomes more compact and, consequently, $d_{\mathrm{bb}}$ increases, while the fractal dimension of the shortest path (see Fig. 10) decreases [6]. For uncorrelated disorder $H=-1$, we measure $d_{\mathrm{bb}}=1.64\pm 0.02$, which is compatible with the results reported in Refs. [66, 87, 89, 92, 93, 95]. The solid line is the graph of the following interpolation: $d_{\mathrm{bb}}\left(H\right)=39/20(1+H)-166/101H$. The inset shows the backbone size as a function of the lattice size $L$ for the same values of $H$ as in the main plot.

Figure 12

Fractal dimension of the red sites $d_{\mathrm{RS}}$ as a function of the Hurst exponent $H$. Based on the result by Coniglio [38, 96, 97], the data (squares) are compared to the theoretical prediction for $1/{\nu }_H$ as a function of $H$, where ${\nu }_H$ is the correlation-length critical exponent of two-dimensional percolation: for $H<-1/{\nu }_{2\mathrm{D}}^{\mathrm{uncorr}}$, $1/{\nu }_H=1/{\nu }_{2\mathrm{D}}^{\mathrm{uncorr}}=3/4$ and for $-1/{\nu }_{2\mathrm{D}}^{\mathrm{uncorr}}\le H<0$, $1/{\nu }_H=-H$ [5, 7, 23]. We note that these results are similar to measurements in Refs. [6, 7, 16, 47, 98]. The inset shows the number of red sites as a function of the lattice size $L$ for the values of $H$ shown in the main plot.

Figure 13

Reduced conductivity exponent $t_H/{\nu }_H$ as a function of the Hurst exponent $H$. For increasing value of $H$, as the backbone becomes more compact (see Fig. 11), $t_H/{\nu }_H$ decreases. For uncorrelated disorder, we find $t_H/{\nu }_H(-1)=-0.992\pm 0.027$, in agreement with Ref. [66]. The solid line corresponds to the expression $t_H/{\nu }_H=16/41-H-7H^2/25$ in the range $-1/{\nu }_{2\mathrm{D}}^{\mathrm{uncorr}}\le H\le 0$ and $t_H/{\nu }_H=t/{\nu }_{2\mathrm{D}}^{\mathrm{uncorr}}$ for $-1\le H\le -1/{\nu }_{2\mathrm{D}}^{\mathrm{uncorr}}$. The inset shows the conductivity $C$ as a function of the lattice size $L$ for the same values of the Hurst exponent $H$ as in the main plot.

Figure 14

Number of bridge sites $M_{\mathrm{br}}$ as a function of the control parameter $p$, for different values of the Hurst exponent $H$, on a lattice of size $L=4096$. Results are averages over ${10}^4$ samples.

Figure 15

Rescaled number of bridge sites $M_{\mathrm{br}}/L^{d_{\mathrm{br}}}$ as a function of the distance to the percolation threshold $p-p_c$, with $H=-0.85$, for different lattice sizes $L$. Here we use $d_{\mathrm{br}}(-0.85)=1.211$ [30, 42, 107]. The inset shows the rescaled number of bridge sites $M_{\mathrm{br}}{L}^{-1/{\nu }_{2\mathrm{D}}^{\mathrm{uncorr}}}$ as a function of the scaling variable $(p-p_c)L^{\theta }$, with $\theta =0.72$. The solid line is a guide to the eye with slope $0.64$.

Figure 16

Rescaled number of bridge sites $M_{\mathrm{br}}/L^{d_{\mathrm{br}}}$ as a function of $p-p_c$, with $H=-0.1$, for different lattice sizes $L$. The inset shows the data for the largest three $L$, with $M_{\mathrm{br}}/L^{0.3}$, as a function of $(p-p_c)L^{0.3}$.

Figure 17

Number of bridge sites $M_{\mathrm{br}}$, for $H=-0.1$, as a function of the lattice size, for different values of the fraction of occupied sites $p=p_c=0.5$, $0.6$, $0.7$, $0.8$, $0.9$, and unity. The solid lines are guides to the eye. The estimated slopes are indicated on the right-hand side of the figure.