Figure 1

An example of a homogeneous FBC rectangle percolation (produced by our simulation program). Here, the rectangles (olive) are of length $l=1$, width $w=0.25$, and aspect ratio $r=4$. The system (of size $L=4$) is defined by the four bold (black) boundaries. For convenience in employing the Newman-Ziff algorithm, each of the two opposite (left and right) boundaries is represented by $L$ ($=4$ here) vertically oriented intersecting auxiliary rectangles [dashed dotted (blue) rectangles]. The system spans (percolates) if two opposite auxiliary rectangles are connected by the edges of normal rectangles, such as the case in this figure. The whole systems are virtually divided by a number of subcells as shown by the dashed (red) lines. All subcells are of the same side length $\sqrt{l^2+w^2}$, except those in the topmost row or rightmost column. Each rectangle belongs to a subcell in which its center lies. Each auxiliary rectangle belongs to its nearest subcell. Note, in general, subcell side length is larger than the rectangle length.Reuse & Permissions

Figure 2

Plots of $N_{0.5}$($L$) against $L^{-1-1/v}$ for homogeneous rectangle systems with (a) $r=1$, (b) $r=10$, (c) $r=100$, and (d) $r=1000$.Reuse & Permissions

Figure 3

Plots of $N_c$ against $2{\pi }^{-1}{A}_r^{-1}$ for homogeneous rectangle systems with $r$ ranging from 1 to ∞. Note all the systems are measured by their rectangle length, i.e., $l=1$. The simulation data (open squares) are also listed in Table . The dashed (blue) and solid (red) curves are fittings by Eqs. (4) and (5), respectively.Reuse & Permissions

Figure 4

The critical remaining area fraction $p_c$ for homogeneous rectangle systems of various aspect ratios (listed in Table ).Reuse & Permissions

Figure 5

Examples of two heterogeneous systems. (a) A percolating realization ($L=10$) for the binary system of [(1,0.1),0.5;(0.5,0.5),0.5]. (b) $N_{0.5}$($L$) against $L^{-1-1/v}$ for the binary system in (a). (c) A percolating realization ($L=10$) for the ternary system of [(1,0.02),0.3;(0.8,0.4),0.4;(0.6,0.1),0.3]. (d) $N_{0.5}$($L$) against $L^{-1-1/v}$ for the ternary system in (c).Reuse & Permissions

Figure 6

Percolation thresholds obtained from simulations (symbols) and calculations (dashed curves) from Eq. (15) for weakly and strongly correlated binary systems. For each group of systems, the number fractions $x_i$ vary between 0 and 1 at the step of 0.1. For calculations, only the systems {(1,0.1);(0.5,0.5)} and {(1,0.02);(0.5,0.5)} are with α $=$ 2.5 in Eq. (15) and are classified as strongly correlated systems. All the others are weakly corrected systems with α $=$ 1.5. Note, for efficiency, not all $N_c$'s in this plot are given by the thermodynamic limit ($L\to \infty$) values as extrapolated from Eq. (3). For {(1,1);(1,0.01)} and {(1,0.5);(1,0.02)}, $N_c$ is approximated by $N_{0.5}$ ($L=64$), whereas, for {(1,0.5);(0.5,0.5)} and {(1,0.01);(0.6,0.6)}, $N_c$ is approximated by $N_{0.5}$ ($L=128$). Actually, however, with respect to the discussions of heterogeneous systems in this paper, the difference between $N_{0.5}$ ($L⩾64$) and the corresponding thermodynamic limit value is negligible [see Figs. 2, 5b, and 5d].Reuse & Permissions

Figure 7

Percolation thresholds obtained from simulations (open squares) and calculations (curves) from Eq. (15) for a group of ultrastrongly correlated binary systems {(1,0.01);(0.1,0.1)}. For the open squares, from left to right, the number fraction $x_1$ increases from 0.1 to 0.9 at a step of 0.1. Note, for these systems, $A_e\ll A_{\infty }$, and the predictions by Eq. (15) with all α's deviate significantly from the simulations. All $N_c$'s are approximated by $N_{0.5}$ ($L=64$).Reuse & Permissions

Figure 8

Percolation thresholds obtained from simulations (symbols) and calculations (dashed curves) for three groups of ternary systems: (a) {(1,0.5);(0.6,0.4);(0.5,0.5)}, (b) {(1,0.02);(0.8,0.4);(0.6,0.6)}, and (c) {(1,0.01);(0.8,0.4);(0.5,0.5)}. All calculations are from Eq. (16) with $\alpha ={\alpha }^{\prime }=1.5$. In all the plots, along the direction of the arrows, the number fraction $x_2$ increases from 0.1 to 0.9 at a step of 0.1. Note, in ternary systems, $x_1=1-x_2-x_3$. All $N_c$'s are approximated by $N_{0.5}$ ($L=64$).Reuse & Permissions