Critical polynomials in the nonplanar and continuum percolation models

Exact or precise thresholds have been intensively studied since the introduction of the percolation model. Recently, the critical polynomial $P_B(p,L)$ was introduced for planar-lattice percolation models, where $p$ is the occupation probability and $L$ is the linear system size. The solution of $P_B=0$ can reproduce all known exact thresholds and leads to unprecedented estimates for thresholds of unsolved planar-lattice models. In two dimensions, assuming the universality of $P_B$, we use it to study a nonplanar lattice model, i.e., the equivalent-neighbor lattice bond percolation, and the continuum percolation of identical penetrable disks, by Monte Carlo simulations and finite-size scaling analysis. It is found that, in comparison with other quantities, $P_B$ suffers much less from finite-size corrections. As a result, we obtain a series of high-precision thresholds $p_c\left(z\right)$ as a function of coordination number $z$ for equivalent-neighbor percolation with $z$ up to $\mathrm{O}\left({10}^5\right)$ and clearly confirm the asymptotic behavior $z{p}_c-1\sim 1/\sqrt{z}$ for $z\to \infty$. For the continuum percolation model, we surprisingly observe that the finite-size correction in $P_B$ is unobservable within uncertainty $\mathrm{O}\left({10}^{-5}\right)$ as long as $L\ge 3$. The estimated threshold number density of disks is ${\rho }_c=1.43632505\left(10\right)$, slightly below the most recent result ${\rho }_c=1.43632545\left(8\right)$ of Mertens and Moore obtained by other means. Our work suggests that the critical polynomial method can be a powerful tool for studying nonplanar and continuum systems in statistical mechanics.

Figure 1

(a) A typical example of a self-dual lattice. The square lattice (solid line) and its dual lattice (dash line) are topologically identical. (b) The asanoha lattice [dual to the $(3,{12}^2)$ lattice]. It is self-matching since any 2D infinite lattice that is fully triangulated is a self-matching lattice. (c) The covering lattice of bond percolation on the square lattice. It is self-matching according to the argument in Ref. [4]. Here the diagonal bonds are nonplanar, and actually for all cases where the faces are not triangular, the matching lattice is always nonplanar.

Figure 2

(a) The star-triangle transformation on the triangular lattice; (b) one individual star-triangle with the bond probabilities $p$ and $p^{*}$.

Figure 3

The triangle-triangle transformation on a basic cell is shown in (a). The shaded region can contain any interactions among the vertices A, B, C. Panel (b) is an example of a self-dual lattice (the bowtie graph) since it is invariant under this triangle-triangle transformation, as shown in (c).

Figure 4

Typical configurations on a 2D square with periodic boundary conditions. The number of different directions along which a configuration wraps decides its type among $\left\{{\mathcal{Z}}_2\right\}$, $\left\{{\mathcal{Z}}_1\right\}$, and $\left\{{\mathcal{Z}}_0\right\}$.

Figure 5

(a) The basic cell of the square lattice with the bond probability $p$. (b) The basic cell of the kagome lattice with the site probability $q$.

Figure 6

$P_B$ versus $L^{-3}$ for bond percolation on a periodic square lattice with $z=8$ equivalent neighbors, at $p=0.250368385$ which is within the error margin of the estimate $p_c=0.25036840\left(4\right)$. The solid line is a straight line with slope $b_1\simeq -0.17$ obtained by fitting the data. From right (small) to left (large), sizes for other data points are $L=9,10,12,14,16,20$, respectively.

Figure 7

Plots of $P_B$, $R_2$, and $Q$ versus $zp$ for different system sizes $L$, for equivalent-neighbor percolation models with $r=15$, corresponding to $z=708$ and $z=960$ for type-1 (left panel) and type-2 (right panel) models, respectively. Vertical dashed lines show the thresholds $z{p}_c=1.102812\left(3\right)$ and $z{p}_c=1.085839\left(5\right)$ [33] for type-1 and type-2 models, respectively. Horizontal dashed lines indicate the universal values of these quantities at criticality. The error bars of the data are smaller than the size of the data points. Values of $L$ are given in the legend. The solid lines connecting data points are added for clarity.

Figure 8

Plots of $P_B$, $R_2$, and $Q$ versus $zp$ for different system sizes $L$, for equivalent-neighbor percolation models with $r=127$, corresponding to $z=50616$ and $z=65024$ for type-1 (left panel) and type-2 (right panel) models, respectively. Vertical dashed lines show the thresholds $z{p}_c=1.011655\left(20\right)$ and $z{p}_c=1.01005\left(3\right)$ [33] for type-1 and type-2 models, respectively. Horizontal dashed lines indicate the universal values of these quantities at criticality. The error bars of the data are smaller than the size of the data points. Values of $L$ are given in the legend. The solid lines connecting data points are added for clarity.

Figure 9

Plot of $(z{p}_c-1)z^{1/2}$ versus $z^{-1/2}$ for equivalent-neighbor models. The straight lines are obtained by fitting the data.

Figure 10

Plots of $P_B$, $R_2$, and $Q$ versus $\rho$ for different system sizes $L$ for the continuum percolation model. Values of $L$ are given in the legend. Vertical dashed line shows the threshold ${\rho }_c=1.43632505\left(10\right)$, and horizontal dashed lines indicate the universal values of these quantities at criticality. The error bars of the data are smaller than the size of the data points. The solid lines connecting data points are added for clarity.

Figure 11

Plot of $P_B$ versus $L$ at different mean densities $\rho$ near criticality. Standard re-weighting technique is applied to obtain the data. Values of $\rho$ are given in the legend. The curves are obtained by fitting the data.

Figure 12

Plot of $P_B$ versus $L$ for $L\le 4$ in continuum percolation, at $\rho =1.4363250$ that is within the error margin of the estimated critical point ${\rho }_c=1.43632505\left(10\right)$. The line connecting data points is added for clarity.