Physical review. B, Condensed matter

Cluster-size statistics of site-bond-correlated percolation models

Paulo R. A. Campos, L.F.C. Pessoa, F. G. Brady Moreira

Published Jul 1, 1997• 14 citations

Abstract

We consider the cluster-size statistics of two types of site-bond-correlated percolation. In model A (B) [see Phys. Rev. B 40, 10 986 (1989)], the activation of a bond between two first-neighbor occupied sites is also dependent on the occupancy of both (either) of their own nearest neighbors, along the line joining them. The critical concentration for models A and B, obtained from simulations on square lattices of sizes up to $2000\ifmmode\times\else\texttimes\fi{}2000$, is found to be $0.7405\ifmmode\pm\else\textpm\fi{}0.0005$ and $0.6145\ifmmode\pm\else\textpm\fi{}0.0005$, respectively. The resulting critical exponents indicate that both models are in the same universality class as ordinary independent (site or bond) percolation.

DOI: 10.1103/physrevb.56.40

Related papers

Two- and three-dimensional site-bond-correlated percolation

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Bond and site percolation in three dimensions

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Physical review. B, Condensed matter

Cluster-size statistics of site-bond-correlated percolation models

Paulo R. A. Campos, L.F.C. Pessoa, F. G. Brady Moreira

Published Jul 1, 1997• 14 citations

Publisher

Abstract

2000\ifmmode\times\else\texttimes\fi{}2000

, is found to be

0.7405\ifmmode\pm\else\textpm\fi{}0.0005

and

0.6145\ifmmode\pm\else\textpm\fi{}0.0005

, respectively. The resulting critical exponents indicate that both models are in the same universality class as ordinary independent (site or bond) percolation.

DOI: 10.1103/physrevb.56.40

Related papers

Two- and three-dimensional site-bond-correlated percolation

N. S. Branco, Kleber D. Machado

We study the site-bond-correlated percolation model in two and three dimensions. In this model, two first-neighbor sites are considered to be joined by an active bond if both are present and if their first neighbors, along the line joining them, are also present. Using a Monte Carlo renormalization-group approach, we calculate percolation thresholds and critical exponents for both square and simple cubic lattices. Our results indicate that the correlated model is in the same universality class as usual percolation, in two and three dimensions; the critical probabilities for the square and simple cubic lattices are found to be 0.7402\ifmmode\pm\else\textpm\fi{}0.0002 and 0.5660\ifmmode\pm\else\textpm\fi{}0.0004, respectively.We study the site-bond-correlated percolation model in two and three dimensions. In this model, two first-neighbor sites are considered to be joined by an active bond if both are present and if their first neighbors, along the line joining them, are also present. Using a Monte Carlo renormalization-group approach, we calculate percolation thresholds and critical exponents for both square and simple cubic lattices. Our results indicate that the correlated model is in the same universality class as usual percolation, in two and three dimensions; the critical probabilities for the square and simple cubic lattices are found to be 0.7402\ifmmode\pm\else\textpm\fi{}0.0002 and 0.5660\ifmmode\pm\else\textpm\fi{}0.0004, respectively.

Physical review. B, Condensed matterJan 1, 199310 citations

Site Percolation in a Square Lattice with Second- and Dilute First-Neighbor Bonds

P.M.C. de Oliveira

A two-parameter real-space renormalization-group approach to the site-percolation problem in a square lattice is proposed. The two parameters are the concentration $p$ of active first-neighbor bonds, and the concentration $q$ of occupied sites. From these two parameters one can calculate two independent critical exponents, and the others can be obtained through scaling relations, with no need of external applied fields.A two-parameter real-space renormalization-group approach to the site-percolation problem in a square lattice is proposed. The two parameters are the concentration $p$ of active first-neighbor bonds, and the concentration $q$ of occupied sites. From these two parameters one can calculate two independent critical exponents, and the others can be obtained through scaling relations, with no need of external applied fields.

Physical Review LettersNov 16, 198111 citations

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M I González, W. Lebrecht, Antonio Jose Ramirez-Pastor

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.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.

Journal of Statistical Mechanics Theory and ExperimentSep 1, 201611 citations

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Songsong Wang, Wanzhou Zhang, Chengxiang Ding

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 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.

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Percolation, clusters, and phase transitions in spin models

Chin‐Kun Hu

The phase transition in the Ising model and the percolation transition in the lattice percolation model have many common characteristics which have motivated researchers to explore whether the former is a percolation transition of a correlated percolation model. Previous attempts to draw such a connection have been either unsuccessful or unsatisfactory. Considering each lattice site with an Ising spin occupied and the nearest-neighbor (NN) coupling between occupied sites as a bond with a bond probability $p$ depending on the NN coupling constant $J$ and the temperature $T$ , we formally show that the partition function of the Ising model is the generating function of bond-correlated percolation model (BCPM) with a bond probability $p=1\ensuremath{-}\mathrm{exp}(\frac{\ensuremath{-}2J}{\mathrm{kT}})$. The BCPM has the Ising critical temperature and exponents, including $\ensuremath{\nu},{\ensuremath{\nu}}^{\ensuremath{'}},\ensuremath{\eta},\ensuremath{\beta},\ensuremath{\alpha},{\ensuremath{\alpha}}^{\ensuremath{'}}$, and $\ensuremath{\gamma}$ (perhaps also ${\ensuremath{\gamma}}^{\ensuremath{'}}$). From the connection between the Ising model and the BCPM, we also derive and hence give a geometrical meaning of the finite-size scaling and broadening at first-order phase transitions of the Ising model. Our approach may easily be extended to many spin models and give geometrical meaning to other properties of spin models.The phase transition in the Ising model and the percolation transition in the lattice percolation model have many common characteristics which have motivated researchers to explore whether the former is a percolation transition of a correlated percolation model. Previous attempts to draw such a connection have been either unsuccessful or unsatisfactory. Considering each lattice site with an Ising spin occupied and the nearest-neighbor (NN) coupling between occupied sites as a bond with a bond probability $p$ depending on the NN coupling constant $J$ and the temperature $T$, we formally show that the partition function of the Ising model is the generating function of bond-correlated percolation model (BCPM) with a bond probability $p=1\ensuremath{-}\mathrm{exp}(\frac{\ensuremath{-}2J}{\mathrm{kT}})$. The BCPM has the Ising critical temperature and exponents, including $\ensuremath{\nu},{\ensuremath{\nu}}^{\ensuremath{'}},\ensuremath{\eta},\ensuremath{\beta},\ensuremath{\alpha},{\ensuremath{\alpha}}^{\ensuremath{'}}$, and $\ensuremath{\gamma}$ (perhaps also ${\ensuremath{\gamma}}^{\ensuremath{'}}$). From the connection between the Ising model and the BCPM, we also derive and hence give a geometrical meaning of the finite-size scaling and broadening at first-order phase transitions of the Ising model. Our approach may easily be extended to many spin models and give geometrical meaning to other properties of spin models.

Physical review. B, Condensed matterMay 1, 1984146 citations

Bond and site percolation in three dimensions

Junfeng Wang, Zongzheng Zhou, Wei Zhang +2 more

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

arXiv:1302.0421Physical Review EMay 7, 2013170 citations

2000\ifmmode\times\else\texttimes\fi{}2000

, is found to be

0.7405\ifmmode\pm\else\textpm\fi{}0.0005

and

0.6145\ifmmode\pm\else\textpm\fi{}0.0005

, respectively. The resulting critical exponents indicate that both models are in the same universality class as ordinary independent (site or bond) percolation.

A two-parameter real-space renormalization-group approach to the site-percolation problem in a square lattice is proposed. The two parameters are the concentration

p

of active first-neighbor bonds, and the concentration

q

of occupied sites. From these two parameters one can calculate two independent critical exponents, and the others can be obtained through scaling relations, with no need of external applied fields.

p

depending on the NN coupling constant

J

and the temperature

T

, we formally show that the partition function of the Ising model is the generating function of bond-correlated percolation model (BCPM) with a bond probability $p=1\ensuremath{-}\mathrm{exp}(\frac{\ensuremath{-}2J}{\mathrm{kT}})$. The BCPM has the Ising critical temperature and exponents, including $\ensuremath{\nu},{\ensuremath{\nu}}^{\ensuremath{'}},\ensuremath{\eta},\ensuremath{\beta},\ensuremath{\alpha},{\ensuremath{\alpha}}^{\ensuremath{'}}$, and $\ensuremath{\gamma}$ (perhaps also ${\ensuremath{\gamma}}^{\ensuremath{'}}$). From the connection between the Ising model and the BCPM, we also derive and hence give a geometrical meaning of the finite-size scaling and broadening at first-order phase transitions of the Ising model. Our approach may easily be extended to many spin models and give geometrical meaning to other properties of spin models.

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 ${y}_{i}\ensuremath{\approx}\ensuremath{-}1.2$.