Extended-range percolation in five dimensions

TL;DR

This work investigates bond and site percolation on the five-dimensional sc(5) lattice with extended neighborhoods, using a high-statistics single-cluster growth Monte Carlo approach to determine the universal exponents $τ$ and $Ω$ and precise $p_c$ thresholds for up to the 7th nearest neighbor. The authors establish a self-consistent method to extract $p_c$, $τ$, and $Ω$ from $s^{τ-2}P_{ geq s}$ and $Q(s)$ plots, finding $τ=2.4177(3)$ and $Ω=0.27(2)$, in agreement with recent five-loop RG predictions. For bond percolation, $p_c$ values confirm Bethe-lattice scaling with finite-$z$ corrections $zp_c-1 \sim z^{-0.88}$ or $(3+\ln z)/z$; for site percolation, $p_c$ follows $p_c \approx c/(z+b)$ with $c=1.722(7)$ and $b=1$, approaching the continuum limit $zp_c \to 32 η_c = 1.742(2)$ as $z$ grows. These results bridge high-dimensional percolation theory with RG and continuum percolation, and quantify finite-size and neighborhood-shape effects in five dimensions.

Abstract

Percolation on a five-dimensional simple hypercubic (sc(5)) lattice with extended neighborhoods is investigated by means of extensive Monte Carlo simulations, using an effective single-cluster growth algorithm. The critical exponents, including $τ$ and $Ω$, the asymptotic behavior of the threshold $p_c$ and its dependence on coordination number $z$ are investigated. Using the bond and site percolation thresholds $p_c = 0.11817145(3)$ and $0.14079633(4)$ respectively given by Mertens and Moore [Phys. Rev. E 98, 022120 (2018)], we find critical exponents of $τ= 2.4177(3)$, $Ω= 0.27(2)$ through a self-consistent process. The value of $τ$ compares favorably with a recent five-loop renormalization predictions $2.4175(2)$ by Borinsky et al. [Phys. Rev. D 103, 116024 (2021)], the value 2.4180(6) that follows from the work of Zhang et al. [Physica A 580, 126124 (2021)], and the measurement of $2.419(1)$ by Mertens and Moore. We also confirmed the bond threshold, finding $p_c = 0.11817150(5)$. sc(5) lattices with extended neighborhoods up to 7th nearest neighbors are studied for both bond and site percolation. Employing the values of $τ$ and $Ω$ mentioned above, thresholds are found to high precision. For bond percolation, the asymptotic value of $zp_c$ tends to Bethe-lattice behavior ($z p_c \sim 1$), and the finite-$z$ correction is found to be consistent with both and $zp_{c} - 1 \sim a_1 z^{-0.88}$ and $zp_{c} - 1 \sim a_0(3 + \ln z)/z$. For site percolation, the asymptotic analysis is close to the predicted behavior $zp_c \sim 32η_c = 1.742(2)$ for large $z$, where $η_c = 0.05443(7)$ is the continuum percolation threshold of five-dimensional hyperspheres given by Torquato and Jiao [J. Chem. Phys 137, 074106 (2015)]; finite-$z$ corrections are accounted for by taking $p_c \approx c/(z + b)$ with $c=1.722(7)$ and $b=1$.

Figures (7)

• Figure 1: Plots of $s^{\tau-2}P_{\geq s}$ with $\tau = 2.4178$ (a) and $Q(s)$ (b) versus $s^{-\Omega}$ with $\Omega = 0.28$ for bond percolation of the sc(5) lattice under different values of $p$ (a), and for $p = 0.11817145$ (b). In (a), we added the linear function $0.024 s^{-\Omega}$ in order to make the slope on the right nearly horizontal and allow us to expand the vertical scale and separate the curves. For $p \ne p_c$, the data in (a) behaves as $0.921+C_1(p-p_c)(s^{-\Omega})^{-\sigma/\Omega}$ with $\sigma/\Omega\approx -1.83$. Data fitting shows that these values of $p_c$, $\tau$ and $\Omega$ are self-consistent.
• Figure 2: Plots of $s^{\tau-2}P_{\geq s}$ with $\tau = 2.4176$ (a) and $Q(s)$ (b) versus $s^{-\Omega}$ with $\Omega = 0.21$ and $\Omega = 0.27$ for site percolation on the sc(5) lattice under $p = 0.14079633$. Clearly, better linear behavior obtains with $\Omega=0.27$ than $\Omega = 0.21$; dashed lines are included as guides to the eye. Data fitting shows that these values of $p_c$, $\tau$ and $\Omega=0.27$ are self-consistent.
• Figure 3: Plot of $s^{\tau-2}P_{\geq s}$ versus $s^{-\Omega}$ with $\tau = 2.4177$ and $\Omega = 0.27$ for bond percolation of the sc(5)-1,2 (a), sc(5)-1,2,3 (b), sc(5)-1,2,3,4 (c), sc(5)-1,...,5 (d), sc(5)-1,...,6 (e), and sc(5)-1,...,7 (f) lattices under different values of $p$.
• Figure 4: Plot of $zp_c$ versus $z^{-x}$ and versus $(3 + \ln z)/z$ for bond percolation on the sc(5)-$1,...,n$ lattices with compact nearest neighborhoods for $n = 3,4,5,6,7$. If we pick $x=0.88$, we get an intercept close to 1, as well as a good linear fit. This data can also evidently be fitted with $zp_c -1 \sim a_1(C + \ln z)/z$ with $C=3$.
• Figure 5: Plots of $s^{\tau-2}P_{\geq s}$ versus $s^{-\Omega}$ with $\tau = 2.4177$ and $\Omega = 0.27$ for site percolation on the lattices sc(5)-1,2 (a), sc(5)-1,2,3 (b), sc(5)-1,2,3,4 (c), sc(5)-1,...,5 (d), sc(5)-1,...,6 (e), and sc(5)-1,...,7 (f) under different values of $p$.