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Physical Review E

Monte Carlo study of the site-percolation model in two and three dimensions

Youjin Deng, Henk W. J. Blöte

Published Jul 21, 2005• 119 citations
Publisher

Abstract

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 $L$ 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 ${p}_{c}=0.592\phantom{\rule{0.2em}{0ex}}746\phantom{\rule{0.2em}{0ex}}5(4)$, and determine two universal quantities as ${Q}_{gbc}=0.930\phantom{\rule{0.2em}{0ex}}34(1)$ and ${Q}_{gsc}=0.793\phantom{\rule{0.2em}{0ex}}72(3)$, 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 ${p}_{c}=0.311\phantom{\rule{0.2em}{0ex}}607\phantom{\rule{0.2em}{0ex}}7(4)$. We further determine the bulk thermal and magnetic exponents as ${y}_{t}=1.1437(6)$ and ${y}_{h}=2.5219(2)$, respectively, and the surface magnetic exponent at the ordinary phase transition as ${y}_{hs}^{(\mathrm{o})}=1.0248(3)$.

DOI
10.1103/physreve.72.016126

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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 Ld\ensuremath−1\ifmmode×\else\texttimes\fi\ensuremath∞{L}^{d\ensuremath{-}1}\ifmmode\times\else\texttimes\fi{}\ensuremath{\infty}Ld\ensuremath−1\ifmmode×\else\texttimes\fi\ensuremath∞, 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{2}^{\ensuremath{-}5∕48}2\ensuremath−5∕48 and 2\ensuremath−1∕3{2}^{\ensuremath{-}1∕3}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).
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Using a histogram Monte Carlo simulation method (HMCSM), Hu, Lin, and Chen found that bond and site percolation models on planar lattices have universal finite-size scaling functions for the existence probability Ep,{E}_{p},Ep​, the percolation probability P,P,P, and the probability Wn{W}_{n}Wn​ for the appearance of nnn percolating clusters in these models. In this paper we extend above study to percolation on three-dimensional lattices with various linear dimensions L.L.L. Using the HMCSM, we calculate the existence probability Ep{E}_{p}Ep​ and the percolation probability PPP for site and bond percolation on a simple-cubic (sc) lattice, and site percolation on body-centered-cubic and face-centered-cubic lattices; each lattice has the same linear dimension in three dimensions. Using the data of Ep{E}_{p}Ep​ and PPP in a percolation renormalization group method, we find that the critical exponents obtained are quite consistent with the universality of critical exponents. Using a small number of nonuniversal metric factors, we find that Ep{E}_{p}Ep​ and PPP have universal finite-size scaling functions. This implies that the critical Ep{E}_{p}Ep​ is a universal quantity, which is $0.265\ifmmode\pm\else\textpm\fi{}0.005$ for free boundary conditions and $0.924\ifmmode\pm\else\textpm\fi{}0.005$ for periodic boundary conditions. We also find that Wn{W}_{n}Wn​ for site and bond percolation on sc lattices have universal finite-size scaling functions.
We investigate bond- and site-percolation models on several two-dimensional lattices numerically, by means of transfer-matrix calculations and Monte Carlo simulations. The lattices include the square, triangular, honeycomb kagome, and diced lattices with nearest-neighbor bonds, and the square lattice with nearest- and next-nearest-neighbor bonds. Results are presented for the bond-percolation thresholds of the kagome and diced lattices, and the site-percolation thresholds of the square, honeycomb, and diced lattices. We also include the bond- and site-percolation thresholds for the square lattice with nearest- and next-nearest-neighbor bonds. We find that corrections to scaling behave according to the second temperature dimension X_{t2}=4 predicted by the Coulomb gas theory and the theory of conformal invariance. In several cases there is evidence for an additional term with the same exponent, but modified by a logarithmic factor. Only for the site-percolation problem on the triangular lattice does such a logarithmic term appear to be small or absent. The amplitude of the power-law correction associated with X_{t2}=4 is found to be dependent on the orientation of the lattice with respect to the cylindrical geometry of the finite systems.
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 $1/\ensuremath{\nu}=1.1410(15)$, $\ensuremath{\beta}/\ensuremath{\nu}=0.477\phantom{\rule{0.16em}{0ex}}05(15)$, the leading correction exponent ${y}_{i}=\ensuremath{-}1.2(2)$, and the shortest-path exponent 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 ${y}_{i}\ensuremath{\approx}\ensuremath{-}1.2$.
In this work we apply a highly efficient Monte Carlo algorithm recently proposed by Newman and Ziff to treat percolation problems. The site and bond percolations are studied on a number of lattices in two and three dimensions. Quite good results for the wrapping probabilities, correlation length critical exponent, and critical concentration are obtained for the square, simple cubic, hexagonal close packed, and hexagonal lattices by using relatively small systems. We also confirm the universal aspect of the wrapping probabilities regarding site and bond dilution.
Using Monte Carlo methods and finite-size scaling, we investigate surface criticality in the O(n)\mathrm{O}(n)O(n) models on the simple-cubic lattice with n=1n=1n=1, 2, and 3, i.e., the Ising, XYXYXY, and Heisenberg models. For the critical couplings we find Kc(n=2)=0.4541659{K}_{\mathrm{c}}(n=2)=0.454\phantom{\rule{0.2em}{0ex}}1659Kc​(n=2)=0.4541659 (10) and Kc(n=3)=0.693003{K}_{\mathrm{c}}(n=3)=0.693\phantom{\rule{0.2em}{0ex}}003Kc​(n=3)=0.693003 (2). We simulate the three models with open surfaces and determine the surface magnetic exponents at the ordinary transition to be yh1(o)=0.7374(15){y}_{h1}^{(\mathrm{o})}=0.7374\phantom{\rule{0.2em}{0ex}}(15)yh1(o)​=0.7374(15), 0.781 (2), and 0.813 (2) for n=1n=1n=1, 2, and 3, respectively. Then we vary the surface coupling K1{K}_{1}K1​ and locate the so-called special transition at ${\ensuremath{\kappa}}_{\mathrm{c}}(n=1)=0.502\phantom{\rule{0.2em}{0ex}}14$ (8) and ${\ensuremath{\kappa}}_{\mathrm{c}}(n=2)=0.6222$ (3), where $\ensuremath{\kappa}={K}_{1}∕K\ensuremath{-}1$. The corresponding surface thermal and magnetic exponents are yt1(s)=0.715{y}_{t1}^{(\mathrm{s})}=0.715yt1(s)​=0.715 (1) and yh1(s)=1.636{y}_{h1}^{(\mathrm{s})}=1.636yh1(s)​=1.636 (1) for the Ising model, and yt1(s)=0.608{y}_{t1}^{(\mathrm{s})}=0.608yt1(s)​=0.608 (4) and yh1(s)=1.675{y}_{h1}^{(\mathrm{s})}=1.675yh1(s)​=1.675 (1) for the XYXYXY model. Finite-size corrections with an exponent close to $\ensuremath{-}1∕2$ occur for both models. Also for the Heisenberg model we find substantial evidence for the existence of a special surface transition.