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arXiv:1302.0421 · Physical Review E

Bond and site percolation in three dimensions

Junfeng Wang, Zongzheng Zhou, Wei Zhang, Timothy M. Garoni, Youjin Deng

Published May 7, 2013• 170 citations
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Abstract

We simulate the bond and site percolation models on a simple-cubic lattice with linear sizes up to $L=512$, and estimate the percolation thresholds to be ${p}_{c}(\mathrm{bond})=0.248\phantom{\rule{0.16em}{0ex}}811\phantom{\rule{0.16em}{0ex}}82(10)$ and ${p}_{c}(\mathrm{site})=0.311\phantom{\rule{0.16em}{0ex}}607\phantom{\rule{0.16em}{0ex}}7(2)$. By performing extensive simulations at these estimated critical points, we then estimate the critical exponents $1/\ensuremath{\nu}=1.1410(15)$, $\ensuremath{\beta}/\ensuremath{\nu}=0.477\phantom{\rule{0.16em}{0ex}}05(15)$, the leading correction exponent ${y}_{i}=\ensuremath{-}1.2(2)$, and the shortest-path exponent ${d}_{\mathrm{min}}=1.3756(3)$. Various universal amplitudes are also obtained, including wrapping probabilities, ratios associated with the cluster-size distribution, and the excess cluster number. We observe that the leading finite-size corrections in certain wrapping probabilities are governed by an exponent $\ensuremath{\approx}$$\ensuremath{-}2$, rather than ${y}_{i}\ensuremath{\approx}\ensuremath{-}1.2$.

DOI
10.1103/physreve.87.052107

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We simulate the bond and site percolation models on a simple-cubic lattice with linear sizes up to L=512L=512L=512, and estimate the percolation thresholds to be pc(bond)=0.24881182(10){p}_{c}(\mathrm{bond})=0.248\phantom{\rule{0.16em}{0ex}}811\phantom{\rule{0.16em}{0ex}}82(10)pc​(bond)=0.24881182(10) and pc(site)=0.3116077(2){p}_{c}(\mathrm{site})=0.311\phantom{\rule{0.16em}{0ex}}607\phantom{\rule{0.16em}{0ex}}7(2)pc​(site)=0.3116077(2). By performing extensive simulations at these estimated critical points, we then estimate the critical exponents 1/\ensuremathν=1.1410(15)1/\ensuremath{\nu}=1.1410(15)1/\ensuremathν=1.1410(15), \ensuremathβ/\ensuremathν=0.47705(15)\ensuremath{\beta}/\ensuremath{\nu}=0.477\phantom{\rule{0.16em}{0ex}}05(15)\ensuremathβ/\ensuremathν=0.47705(15), the leading correction exponent yi=\ensuremath−1.2(2){y}_{i}=\ensuremath{-}1.2(2)yi​=\ensuremath−1.2(2), and the shortest-path exponent dmin=1.3756(3){d}_{\mathrm{min}}=1.3756(3)dmin​=1.3756(3). Various universal amplitudes are also obtained, including wrapping probabilities, ratios associated with the cluster-size distribution, and the excess cluster number. We observe that the leading finite-size corrections in certain wrapping probabilities are governed by an exponent \ensuremath{\approx}$$\ensuremath{-}2, rather than yi\ensuremath≈\ensuremath−1.2{y}_{i}\ensuremath{\approx}\ensuremath{-}1.2yi​\ensuremath≈\ensuremath−1.2.
We investigate the site-percolation problem on the square and simple-cubic lattices by means of a Monte Carlo algorithm that in fact simulates systems with size ${L}^{d\ensuremath{-}1}\ifmmode\times\else\texttimes\fi{}\ensuremath{\infty}$, where LLL specifies the linear system size. This algorithm can be regarded either as an extension of the Hoshen-Kopelman method or as a special case of the transfer-matrix Monte Carlo technique. Various quantities, such as the magnetic correlation function, are sampled in the finite directions of the above geometry. Simulations are arranged such that both bulk and surface quantities can be sampled. On the square lattice, we locate the percolation threshold at pc=0.5927465(4){p}_{c}=0.592\phantom{\rule{0.2em}{0ex}}746\phantom{\rule{0.2em}{0ex}}5(4)pc​=0.5927465(4), and determine two universal quantities as Qgbc=0.93034(1){Q}_{gbc}=0.930\phantom{\rule{0.2em}{0ex}}34(1)Qgbc​=0.93034(1) and Qgsc=0.79372(3){Q}_{gsc}=0.793\phantom{\rule{0.2em}{0ex}}72(3)Qgsc​=0.79372(3), which are associated with bulk and surface correlations, respectively. These values agree well with the exact values ${2}^{\ensuremath{-}5∕48}$ and ${2}^{\ensuremath{-}1∕3}$, respectively, which follow from conformal invariance. On the simple-cubic lattice, we locate the percolation threshold at pc=0.3116077(4){p}_{c}=0.311\phantom{\rule{0.2em}{0ex}}607\phantom{\rule{0.2em}{0ex}}7(4)pc​=0.3116077(4). We further determine the bulk thermal and magnetic exponents as yt=1.1437(6){y}_{t}=1.1437(6)yt​=1.1437(6) and yh=2.5219(2){y}_{h}=2.5219(2)yh​=2.5219(2), respectively, and the surface magnetic exponent at the ordinary phase transition as yhs(o)=1.0248(3){y}_{hs}^{(\mathrm{o})}=1.0248(3)yhs(o)​=1.0248(3).
Abstract The site-percolation problem on simple cubic lattices has been studied by means of numerical simulation and analytical calculations based on exact counting of configurations on finite cells. Motivated by considerations of cluster connectivity, two distinct schemes (denoted as S ∩ B and S ∪ B ) have been considered. In S ∩ B ( S ∪ B ), two points are said to be connected if a sequence of occupied sites and ( or ) bonds joins them. Theoretical and numerical results, supplemented by analysis using finite-size scaling theory, were used to calculate the complete phase diagram of the system in the ( p s , p b ) space. Our study allowed us also to determine the critical exponents (and universality) characterizing the phase transition occurring in the system.
We present a Monte Carlo study of the bond- and site-directed (oriented) percolation models in (d+1) dimensions on simple-cubic and body-centered-cubic lattices, with 2 ≤ d ≤ 7. A dimensionless ratio is defined, and an analysis of its finite-size scaling produces improved estimates of percolation thresholds. We also report improved estimates for the standard critical exponents. In addition, we study the probability distributions of the number of wet sites and radius of gyration, for 1 ≤ d ≤ 7.
We propose a site random-cluster model by introducing an additional cluster weight in the partition function of the traditional site percolation. To simulate the model on a square lattice, we combine the color-assignation and the Swendsen-Wang methods to design a highly efficient cluster algorithm with a small critical slowing-down phenomenon. To verify whether or not it is consistent with the bond random-cluster model, we measure several quantities, such as the wrapping probability Re, the percolating cluster density P∞, and the magnetic susceptibility per site χp, as well as two exponents, such as the thermal exponent yt and the fractal dimension yh of the percolating cluster. We find that for different exponents of cluster weight q=1.5, 2, 2.5, 3, 3.5, and 4, the numerical estimation of the exponents yt and yh are consistent with the theoretical values. The universalities of the site random-cluster model and the bond random-cluster model are completely identical. For larger values of q, we find obvious signatures of the first-order percolation transition by the histograms and the hysteresis loops of percolating cluster density and the energy per site. Our results are helpful for the understanding of the percolation of traditional statistical models.
We study the bond-percolation model in two and three dimensions by Monte Carlo simulation, and investigate the finite-size scaling behavior of several quantities that account for fluctuations of the total numbers of clusters and occupied bonds, Nc{N}_{c}Nc​ and Nb{N}_{b}Nb​, respectively. These quantities include ${C}_{2c}=(⟨{N}_{c}^{2}⟩\ensuremath{-}{⟨{N}_{c}⟩}^{2})∕{L}^{d}$ and ${C}_{cb}=(⟨{N}_{c}{N}_{b}⟩\ensuremath{-}⟨{N}_{c}⟩⟨{N}_{b}⟩)∕{L}^{d}$, where LLL is the linear system size and ddd is the spatial dimensionality. In statistical models with thermal fluctuations, C2c{C}_{2c}C2c​ and Ccb{C}_{cb}Ccb​ are specific heatlike quantities. Despite the absence of thermal fluctuations in percolation, we find that the leading finite-size scaling of C2c{C}_{2c}C2c​ and Ccb{C}_{cb}Ccb​ is described by the thermal critical exponent ${y}_{t}\text{\ensuremath{-}}d$. We also measure quantity ${\ensuremath{\kappa}}_{b}=2⟨{N}_{b}{S}_{2}⟩∕⟨{S}_{2}⟩\ensuremath{-}⟨{N}_{b}{\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{S}}_{4}⟩∕⟨{\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{S}}_{4}⟩\ensuremath{-}⟨{N}_{b}⟩$ and an analogous quantity ${\ensuremath{\kappa}}_{c}$ for Nc{N}_{c}Nc​, where S2{S}_{2}S2​ and ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{S}}_{4}$ are quantities associated with the second and the fourth moments of cluster sizes, respectively. At criticality, we show that ${\ensuremath{\kappa}}_{b}$ and ${\ensuremath{\kappa}}_{c}$ diverge as Lyt{L}^{{y}_{t}}Lyt​ for $L\ensuremath{\rightarrow}\ensuremath{\infty}$. The analysis of the data of ${\ensuremath{\kappa}}_{b}$ and ${\ensuremath{\kappa}}_{c}$ yields yt=1.145(2){y}_{t}=1.145(2)yt​=1.145(2) for the three-dimensional percolation, in good agreement with existing results.
We study the cubic fixed point for N=3N=3N=3 and 4 by using finite-size scaling applied to data obtained from Monte Carlo simulations of the NNN-component ${\ensuremath{\phi}}^{4}$ model on the simple cubic lattice. We generalize the idea of improved models to a two-parameter family of models. The two-parameter space is scanned for the point, where the amplitudes of the two leading corrections to scaling vanish. To this end, a dimensionless quantity is introduced that monitors the breaking of the O(N)O(N)O(N) invariance. For N=4N=4N=4, we determine the correction exponents ${\ensuremath{\omega}}_{1}=0.763(24)$ and ${\ensuremath{\omega}}_{2}=0.082(5)$. In the case of N=3N=3N=3, we obtain Y4=0.0142(6){Y}_{4}=0.0142(6)Y4​=0.0142(6) for the renormalization group exponent of the cubic perturbation at the O(3)O(3)O(3)-invariant fixed point, while the correction exponent ${\ensuremath{\omega}}_{2}=0.0133(8)$ at the cubic fixed point. Simulations close to the improved point result in the estimates $\ensuremath{\nu}=0.7202(7)$ and $\ensuremath{\eta}=0.0371(2)$ of the critical exponents of the cubic fixed point for N=4N=4N=4. For N=3N=3N=3, at the cubic fixed point, the O(3)O(3)O(3) symmetry is only mildly broken and the critical exponents differ only by little from those of the O(3)O(3)O(3)-invariant fixed point. We find $\ensuremath{-}0.000\phantom{\rule{0.16em}{0ex}}01⪅{\ensuremath{\eta}}_{\text{cubic}}\ensuremath{-}{\ensuremath{\eta}}_{O(3)}⪅0.000\phantom{\rule{0.16em}{0ex}}07$ and ${\ensuremath{\nu}}_{\text{cubic}}\ensuremath{-}{\ensuremath{\nu}}_{O(3)}=\ensuremath{-}0.000\phantom{\rule{0.16em}{0ex}}61(10)$.