Site and bond percolation on four-dimensional simple hypercubic lattices with extended neighborhoods

We use the notation sc(4)-a, b, ... to indicate a four-dimensional simple hypercubic lattice with the ath nearest neighbors, bth nearest neighbors, etc. In the Monte-Carlo simulations, by using a single-cluster growth algorithm (see [24, 25, 33] for more details), a site on the lattice of size L × L × L × L with L = 128 under periodic boundary conditions is chosen as the seed, and an individual cluster is grown from that seeded site. We grow many samples of individual clusters for each lattice, which are 5 × 10⁸ for sc(4)-3, sc(4)-1,3, sc(4)-2,3, sc(4)-1,2, sc(4)-1,2,3, and sc(4)-1,2,3,4 lattices, and 10⁸ for other five lattices. Clusters with different sizes are distributed in bins of range of (2ⁿ , 2ⁿ⁺¹ − 1) for n = 0, 1, 2, .... An upper size of the cluster, which is called the upper size cutoff, needs to be set to halt the growth of clusters larger than that size, to avoid wrapping around the boundaries. For clusters still growing when they reach an upper size cutoff, we count them in the last bin. Here for site percolation, we set the upper size cutoff to be 2¹⁵ occupied sites for the sc(4)-1,2 and sc(4)-1,2,3 lattices, 2¹³ for the sc(4)-1, ..., 9 lattice, and 2¹⁴ for other lattices. For bond percolation, we set the upper size cutoff to be 2¹⁶ occupied (wetted) sites for all lattices.

If one defines n_s (p) as the number of clusters (per site) containing s occupied sites, as a function of the site or bond occupation probability p, then in the scaling limit, in which s is large and (p − p_c ) is small such that (p − p_c )s^σ is constant, n_s (p) behaves as

$$\begin{equation}{n}_{s}(p)\sim {A}_{0}{s}^{-\tau }f[{B}_{0}(p-{p}_{c}){s}^{\sigma }],\end{equation} \tag{ 7 }$$

where τ, σ, and f(x) are universal, while A₀ and B₀ are lattice-dependent metric factors. When p = p_c and for finite s, there are corrections to equation (7)

$$\begin{equation}{n}_{s}({p}_{c})\sim {A}_{0}{s}^{-\tau }(1+{C}_{0}{s}^{-{\Omega}}+\dots \enspace ),\end{equation} \tag{ 8 }$$

where Ω is another universal exponent. Then the probability that a point belongs to a cluster of size greater than or equal to s is given by ${P}_{\geqslant s}={\sum }_{{s}^{\prime }=s}^{\infty }{s}^{\prime }{n}_{{s}^{\prime }}$, and it follows from equations (7) and (8) that [24, 25]

$$\begin{equation}{s}^{\tau -2}{P}_{\geqslant s}\sim {A}_{1}[1+{B}_{1}(p-{p}_{c}){s}^{\sigma }+{C}_{1}{s}^{-{\Omega}}+\dots \enspace ],\end{equation} \tag{ 9 }$$

where A₁, B₁ and C₁ are non-universal constants. Equation (9) provides two methods to determine the percolation threshold p_c , and we will show them in detail in the next subsection, combined with our simulation results.

Due to the universality of τ, Ω and σ, these exponents depend only on the system dimension, not the type of the lattice. We choose the central values of the estimates τ = 2.3135(5) [24], Ω = 0.40(3) [24], and σ = 0.4742 [34, 35], which are relatively accurate and acceptable, in the numerical simulations. The number of clusters greater than or equal to size s could be found based on the data from our simulation, and the quantity s^τ−2 P_⩾s could be easily calculated.

From the behavior of equation (9), we can determine if we are above, near, or below the percolation threshold. For large s where the finite-size effect term s^−Ω can be ignored, equation (9) becomes

$$\begin{equation}{s}^{\tau -2}{P}_{\geqslant s}\sim {A}_{1}[1+{B}_{1}(p-{p}_{c}){s}^{\sigma }].\end{equation} \tag{ 10 }$$

This implies that s^τ−2 P_⩾s will convergence to a constant value at p_c , while it deviates linearly from that constant value when p is away from p_c , when plotted as a function of s^σ . Figure 1 shows the relation of s^τ−2 P_⩾s versus s^σ for the site percolation of the sc(4)-1, ..., 9 lattice under probabilities p = 0.004 828, 0.004 829, 0.004 830, 0.004 831, 0.004 832, and 0.004 833. For small clusters, s^τ−2 P_⩾s shows a steep rise due to the finite-size effect, while for large clusters, s^τ−2 P_⩾s shows a linear region. As p tends to p_c , the linear part of s^τ−2 P_⩾s become more nearly horizontal. Based upon these properties of the linear portions in figure 1, the central value of p_c can be determined from the slope:

$$\begin{equation}\frac{\mathrm{d}({s}^{\tau -2}{P}_{\geqslant s})}{\mathrm{d}({s}^{\sigma })}\sim {B}_{1}(p-{p}_{c}),\end{equation} \tag{ 11 }$$

As shown in the inset of figure 1, p_c = 0.004 8301 can be deduced from the p intercept of the plot of the above derivative versus p.

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If p is very close to p_c , equation (9) reduces to

$$\begin{equation}{s}^{\tau -2}{P}_{\geqslant s}\sim {A}_{1}[1+{C}_{1}{s}^{-{\Omega}}],\end{equation} \tag{ 12 }$$

implying a linear relationship between s^τ−2 P_⩾s and s^−Ω for large s. In figure 2, we show the plot of s^τ−2 P_⩾s versus s^−Ω for the site percolation of the sc(4)-1, ..., 9 lattice under probabilities p = 0.004 828, 0.004 829, 0.004 830, 0.004 831, 0.004 832, and 0.004 833. It can be seen that when p is away from p_c , the curves show an obvious deviation from linearity for large s, while better linear behavior emerges if p is very close to p_c . Then we can conclude the range of the site percolation threshold of 0.004 830 < p_c < 0.004 831 here.

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Based upon the two methods indicated in figures 1 and 2, as well as the errors for the values of τ = 2.3135(5) and Ω = 0.40(3), finally, the site percolation threshold of the sc(4)-1, ... ,9 lattice can be deduced to be p_c = 0.0048301(9), where the number in parentheses represents the estimated error in the last digit. The figures for the other seven lattices we simulated are shown in the supplementary material [36] in figures 1–14 (https://stacks.iop.org/JSTAT/2022/033202/mmedia), and the corresponding site percolation thresholds are summarized in table 1.

In table 1, we also show in the last column previous site percolation thresholds for the sc(4)-1,2 and sc(4)-1,2,3 lattices provided by [19]. For these two lattices, we get substantially more precise values. For the other six lattices, it appears that our results are new.

With regard to the asymptotic behavior between percolation thresholds p_c and coordination number z for site percolation on lattices with compact nearest neighborhoods in four dimensions where η_c for hyperspheres equals 0.1304(5) [37], one should expect from equation (1) that

$$\begin{equation}{p}_{c}=\frac{2.086(8)}{z}.\end{equation} \tag{ 13 }$$

The extended-range nearest-neighbor regions on a lattice are not exactly spherical in shape, of course. A limit of a more non-spherical surface is a system aligned hypercubes, where η_c = 0.1201(6) [37], implying c = zp_c = 16η_c = 1.922(10), which is not much smaller than the value above. Indeed, for some small values of z considered here, the neighborhood tends to a hypercube. However, for large z where the neighborhood becomes more spherical, one would expect that zp_c should approach the value 2.086.

By plotting the relation of z versus 1/p_c and z versus − 1/ln(1 − p_c ), as shown in figures 3 and 4, respectively, we can see that data fitting of both plots lead to the asymptotic value of 2.065(11), which agrees quite well with the theoretically predicted value for hyperspheres above.

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In contrast to the case of two and three dimensions [32], here we find that the plots of z versus 1/p_c and z versus −1/ln(1 − p_c ) have very similar behavior and give an identical slope. This is because the values of z considered here are much larger than we considered in the lower dimensions, and for large z and small p_c the formulas are nearly identical. If we compare the slopes and intercepts of the two plots:

$$\begin{equation}\begin{gathered}z=\frac{c}{{p}_{c}}-b\hspace{25.0pt}\text{and}\hspace{25.0pt}z=\frac{-{c}^{\prime }}{\mathrm{ln}(1-{p}_{c})}-{b}^{\prime }={c}^{\prime }\left(\frac{1}{{p}_{c}}-\frac{1}{2}-\frac{{p}_{c}}{12}\dots \right)-{b}^{\prime }\end{gathered}\end{equation} \tag{ 14 }$$

we see, ignoring $\mathcal{O}({p}_{c})$ and higher-order terms, that the two plots are equivalent, with c' = c and b' = b − c'/2. In fact, our measured values c = c' = 2.065(11), b = 1.290(12) and b' = 0.250(12) agree with these formulas. For systems of large z, the general formula equation (2) is sufficient and for fitting the data, and the formula (3) is not more beneficial.

For the bond percolation simulated in this paper, figures 5 and 6 show the plots of s^τ−2 P_⩾s versus s^σ and s^−Ω, respectively, for the sc(4)-1, ..., 9 lattice under probabilities p = 0.002 4105, 0.002 4110, 0.002 4115, 0.002 4120, 0.002 4125, and 0.002 4130. Similar to the discussion above, we conclude the bond percolation threshold of the sc(4)-1, ..., 9 lattice here to be p_c = 0.0024117(7). The figures for the other 9 lattices we simulated are shown in the supplementary material [36] in figures 15–32, and the corresponding bond percolation thresholds are summarized in table 2.

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Table 2 also shows the values of zp_c for each lattice. With the increase of z, the value of zp_c decreases and tends to the asymptotic value of 1. In figure 7, we show the relation of zp_c versus z^−x with x = 0.5, 0.75, and 0.9, and also versus (ln z)/z. We see that the behavior is consistent with the prediction of equation (5), but we cannot really rule out the behavior of equation (6) with x = 3/4, which follows the behavior x = (d − 1)/d predicted for d = 2 and 3.

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