Standard and inverse bond percolation of straight rigid rods on square lattices

Numerical simulations and finite-size scaling analysis have been carried out to study standard and inverse bond percolation of straight rigid rods on square lattices. In the case of standard percolation, the lattice is initially empty. Then, linear bond $k$-mers (sets of $k$ linear nearest-neighbor bonds) are randomly and sequentially deposited on the lattice. Jamming coverage $p_{j,k}$ and percolation threshold $p_{c,k}$ are determined for a wide range of $k$ ($1\le k\le 120$). $p_{j,k}$ and $p_{c,k}$ exhibit a decreasing behavior with increasing $k$, $p_{j,k\to \infty }=0.7476\left(1\right)$ and $p_{c,k\to \infty }=0.0033\left(9\right)$ being the limit values for large $k$-mer sizes. $p_{j,k}$ is always greater than $p_{c,k}$, and consequently, the percolation phase transition occurs for all values of $k$. In the case of inverse percolation, the process starts with an initial configuration where all lattice bonds are occupied and, given that periodic boundary conditions are used, the opposite sides of the lattice are connected by nearest-neighbor occupied bonds. Then, the system is diluted by randomly removing linear bond $k$-mers from the lattice. The central idea here is based on finding the maximum concentration of occupied bonds (minimum concentration of empty bonds) for which connectivity disappears. This particular value of concentration is called the inverse percolation threshold $p_{c,k}^i$, and determines a geometrical phase transition in the system. On the other hand, the inverse jamming coverage $p_{j,k}^i$ is the coverage of the limit state, in which no more objects can be removed from the lattice due to the absence of linear clusters of nearest-neighbor bonds of appropriate size. It is easy to understand that $p_{j,k}^i=1-p_{j,k}$. The obtained results for $p_{c,k}^i$ show that the inverse percolation threshold is a decreasing function of $k$ in the range $1\le k\le 18$. For $k>18$, all jammed configurations are percolating states, and consequently, there is no nonpercolating phase. In other words, the lattice remains connected even when the highest allowed concentration of removed bonds $p_{j,k}^i$ is reached. In terms of network attacks, this striking behavior indicates that random attacks on single nodes ($k=1$) are much more effective than correlated attacks on groups of close nodes (large $k$'s). Finally, the accurate determination of critical exponents reveals that standard and inverse bond percolation models on square lattices belong to the same universality class as the random percolation, regardless of the size $k$ considered.

Figure 1

(a) Schematic representation of a square lattice, initially empty, in which some linear bond 3-mers have been deposited. Thick (black) lines and thin (blue) lines represent 3-mer units (occupied bonds) and empty (nonoccupied) bonds, respectively. (b) Schematic representation of an initially fully occupied square lattice in which some linear bond 3-mers have been removed. Thick (black) lines and thin (red) lines represent occupied and removed bonds, respectively.

Figure 2

Curves of the jamming probability $W_L$ as a function of the fraction of occupied bonds $p$ for two values of $k$-mer size ($k=2$, curves on the left, and $k=5$, curves on the right) and lattice sizes ranging between $L/k=128$ and $L/k=640$. The statistical error is smaller than the symbol size. Horizontal dashed line shows the $W_L^{*}$ point. Vertical dashed lines denote the jamming thresholds in the thermodynamic limit.

Figure 3

Jamming coverage $p_{j,k}$ as a function of $k$ for linear bond $k$-mers on square lattices with $k$ between 2 and 120. Inset: As main figure for $2\le k\le 40$. Solid squares represent simulation results obtained in this work (the error in each measurement is smaller than the size of the symbols), open triangles denote data from the literature [31], and the solid line corresponds to the fitting function as discussed in the text.

Figure 4

Extrapolation of the percolation threshold for an $L$-lattice $p_{c,k}^X\left(L\right)(X=\{I,U,A\})$ toward the thermodynamic limit according to the theoretical prediction given by Eq. (6). Triangles, circles, and squares denote the values of $p_{c,k}^X\left(L\right)$ obtained by using the criteria $I,A$, and $U$, respectively. Two values of $k$ are presented: (a) $k=2$ and (b) $k=5$. The bar error in each measurement is smaller than the size of the symbols.

Figure 5

Percolation threshold $p_{c,k}$ as a function of $k$ for linear bond $k$-mers on square lattices (solid squares). The behavior of the fitting curve is described according to Eq. (7). The asymptotic limit $p_{c,k\to \infty }=0.0033\left(9\right)$ is shown. The size of the points is larger than the corresponding error bars. Open triangles denote previous data in the literature [31].

Figure 6

Extrapolation of the inverse percolation threshold for an $L$-lattice $p_{c,k}^{i,X}\left(L\right)(X=\{I,U,A\})$ toward the thermodynamic limit according to the theoretical prediction given by Eq. (6). Triangles, circles, and squares denote the values of $p_{c,k}^{i,X}\left(L\right)$ obtained by using the criteria $I,A$, and $U$, respectively. Two values of $k$ are presented: (a) $k=2$; and (b) $k=5$. The bar error in each measurement is smaller than the size of the symbols.

Figure 7

Inverse percolation threshold $p_{c,k}^i$ (open circles) and limit coverage $p_{j,k}$ (solid circles) as a function of $k$ for linear bond $k$-mers on square lattices. The size of the points is larger than the corresponding error bars. Inset: Zoom of the main figure for $17\le k\le 19$.

Figure 8

Fraction of percolating lattices $R_{L,k}^X$ $[X=I$ (blue curves), $U$ (red curves), $A$ (black curves)] as a function of the concentration $\theta$ for $k=17$ (a), $k=18$ (b), and $k=19$ (c), and three different lattice sizes: $L/k=64$, squares; $L/k=128$, circles; and $L/k=256$, triangles. The statistical error is smaller than the symbol size.

Figure 9

(a) Maximum of the derivative of the $A$ percolation probability ${\left(d{R}_{L,k}^A/dp\right)}_{\max }$ as a function of $L/k$ (in a log-log scale) for four different cases: standard percolation, $k=2$ (solid triangles); standard percolation, $k=5$ (solid squares); inverse percolation, $k=2$ (open triangles); and inverse percolation, $k=5$ (open squares). The bar error in each measurement is smaller than the size of the symbols. According to Eq. (9) the slope of each line corresponds to $1/\nu =3/4$. (b) Natural logarithm of the standard deviation in Eq. (5) $ln\left({\mathrm{\Delta }}_{L,k}^A\right)$ as a function of $L/k$ (in a log-log scale) for the same cases shown in part (a). According to Eq. (10), the slope of each line corresponds to $-1/\nu =-3/4$.

Figure 10

(a) Maximum of the susceptibility ${\chi }_{\max }$ as a function of $L/k$ (in a log-log scale) for the following cases: standard percolation, $k=2$ (solid triangles); standard percolation, $k=5$ (solid squares); inverse percolation, $k=2$ (open triangles); and inverse percolation, $k=5$ (open squares). The bar error in each measurement is smaller than the size of the symbols. The slope of each line corresponds to $\gamma /\nu =43/24$. (b) Maximum of the derivative of the percolation order parameter ${\left(dP/dp\right)}_{\max }$ as a function of $L/k$ (in a log-log scale) for the same cases reported in part (a). According to Eq. (14), the slope of each line corresponds to $(1-\beta )/\nu =31/48$.