Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Journal of Physics C Solid State Physics

A Monte Carlo calculation of the cluster size critical exponent for 2D bond percolation

Christopher J. Lobb, Keith R. Karasek

Published Apr 20, 1980• 13 citations
Publisher

Abstract

Reynolds, Stanley and Klein (1978) developed a large-cell renormalisation group scheme and applied it to the site percolation problem in two dimensions. Applying this technique to bond percolation, the authors calculate the cluster size critical exponent nu . This method converges very rapidly when applied to the bond problem. They estimate that nu =1.343+or-0.017, in excellent agreement with the earlier site percolation result, thus supporting the idea of site-bond universality.

DOI
10.1088/0022-3719/13/11/001

Related papers

Critical exponents for two-dimensional bond percolation

Christopher J. Lobb, Keith R. Karasek

We have determined the critical parameters for bond percolation on the square lattice using the recently developed large-cell renormalization-group method of Reynolds, Stanley, and Klein. The self-dual symmetry of our cells, which they share with the lattice itself, allows considerable simplification of the method and leads to rapid convergence toward the infinite-cell limit. We find that the scaling powers are given by ${y}_{p}=0.750\ifmmode\pm\else\textpm\fi{}0.005$ and ${y}_{h}=1.90\ifmmode\pm\else\textpm\fi{}0.017$. This leads, via scaling, to critical exponents $\ensuremath{\alpha}=\ensuremath{-}0.667\ifmmode\pm\else\textpm\fi{}0.018$, $\ensuremath{\beta}=0.133\ifmmode\pm\else\textpm\fi{}0.023$, $\ensuremath{\gamma}=2.400\ifmmode\pm\else\textpm\fi{}0.048$, $\ensuremath{\delta}=19.0\ifmmode\pm\else\textpm\fi{}3.4$, $\ensuremath{\nu}=1.333\ifmmode\pm\else\textpm\fi{}0.009$, and $\ensuremath{\eta}=0.200\ifmmode\pm\else\textpm\fi{}0.034$.

Physical review. B, Condensed matter•Jan 1, 1982•13 citations

A large-cell renormalisation group calculation of the percolation conduction critical exponent

Christopher J. Lobb, David J. Frank

The authors employ a position-space renormalisation method, using square cells containing up to 365 bonds, to calculate the conduction critical exponent t for the two-dimensional random resistor network problem. Their results are consistent with t being equal to the cluster size critical exponent v.

Journal of Physics C Solid State Physics•Nov 28, 1979•65 citations

A real-space renormalization group for site and bond percolation

PETER J. REYNOLDS, Eugene Stanley, W. Klein

A real-space renormalization group is developed which renormalizes probabilities directly in the percolation problem. An exact transformation is given in one dimension, and a cluster approach is presented for other lattices. The results are in excellent agreement with series calculations for the critical percolation concentration p c (both site and bond), and in good agreement for the correlation length exponent nu p . Additionally, in one dimension, a field-like variable is included and the remaining exponents are calculated.

Journal of Physics C Solid State Physics•Apr 28, 1977•270 citations

Large-cell Monte Carlo renormalization group: A new weight function for percolation

Naeem Jan, D. L. Hunter, Turab Lookman

The large-cell Monte Carlo renormalization group introduced by Reynolds, Stanley, and Klein [Phys. Rev. B 20, 1223 (1980)] is applied with a new weight function to random percolation in two and three dimensions. Our renormalization-group results show different behavior with lattice size than that observed previously. Specifically, we find that ${Y}_{p}=0.73\ifmmode\pm\else\textpm\fi{}0.01$ (the correlation length exponent $\ensuremath{\nu}=1.37$) in two dimensions and ${Y}_{p}=1.124\ifmmode\pm\else\textpm\fi{}0.004$ ($\ensuremath{\nu}=0.89$) in three dimensions. This weight function allows for easy analysis of percolation in higher dimensions.

Physical review. B, Condensed matter•Jun 1, 1984•6 citations

Renormalization Group Approach for Critical Percolation Behavior in a Two Dimensional Bond Percolating Square Lattice

Chiaki Murase

A renormalization group approach in real space is applied to the bond percolating square lattice in two dimension using two scaling transformations proposed by Reynolds et al. (Reynolds's method) and by Yuge (Yuge's method). The renormalization transformation R b ( p ), the eigenvalue λ( b ) and the correlation length critical exponent v ( b ) are calculated exactly by changing the scale of length b =4 for Reynolds's method and \(b=5/\sqrt{2}\) and \(6/\sqrt{2}\) for Yuge's method. The true critical exponent v is estimated v =1.341 and 1.337 by extrapolation, respectively. Discussions are made on the results.

Journal of the Physical Society of Japan•Apr 1, 1981•0 citations

Histogram Monte Carlo renormalization-group method for percolation problems

Chin‐Kun Hu

We present a histogram Monte Carlo method to calculate the existence probability ${\mathit{E}}_{\mathit{p}}$, the percolation probability P, and the mean-cluster size S for percolation problems as continuous functions of the bond or site probability p. We then use ${\mathit{E}}_{\mathit{p}}$ and P in a percolation renormalization-group method to calculate the critical point and exponents. Our method gives quite accurate results for percolation problems.

Physical review. B, Condensed matter•Sep 1, 1992•64 citations
Reynolds, Stanley and Klein (1978) developed a large-cell renormalisation group scheme and applied it to the site percolation problem in two dimensions. Applying this technique to bond percolation, the authors calculate the cluster size critical exponent nu . This method converges very rapidly when applied to the bond problem. They estimate that nu =1.343+or-0.017, in excellent agreement with the earlier site percolation result, thus supporting the idea of site-bond universality.
We have determined the critical parameters for bond percolation on the square lattice using the recently developed large-cell renormalization-group method of Reynolds, Stanley, and Klein. The self-dual symmetry of our cells, which they share with the lattice itself, allows considerable simplification of the method and leads to rapid convergence toward the infinite-cell limit. We find that the scaling powers are given by ${y}_{p}=0.750\ifmmode\pm\else\textpm\fi{}0.005$ and ${y}_{h}=1.90\ifmmode\pm\else\textpm\fi{}0.017$. This leads, via scaling, to critical exponents $\ensuremath{\alpha}=\ensuremath{-}0.667\ifmmode\pm\else\textpm\fi{}0.018$, $\ensuremath{\beta}=0.133\ifmmode\pm\else\textpm\fi{}0.023$, $\ensuremath{\gamma}=2.400\ifmmode\pm\else\textpm\fi{}0.048$, $\ensuremath{\delta}=19.0\ifmmode\pm\else\textpm\fi{}3.4$, $\ensuremath{\nu}=1.333\ifmmode\pm\else\textpm\fi{}0.009$, and $\ensuremath{\eta}=0.200\ifmmode\pm\else\textpm\fi{}0.034$.
A real-space renormalization group is developed which renormalizes probabilities directly in the percolation problem. An exact transformation is given in one dimension, and a cluster approach is presented for other lattices. The results are in excellent agreement with series calculations for the critical percolation concentration p c (both site and bond), and in good agreement for the correlation length exponent nu p . Additionally, in one dimension, a field-like variable is included and the remaining exponents are calculated.
The large-cell Monte Carlo renormalization group introduced by Reynolds, Stanley, and Klein [Phys. Rev. B 20, 1223 (1980)] is applied with a new weight function to random percolation in two and three dimensions. Our renormalization-group results show different behavior with lattice size than that observed previously. Specifically, we find that ${Y}_{p}=0.73\ifmmode\pm\else\textpm\fi{}0.01$ (the correlation length exponent $\ensuremath{\nu}=1.37$) in two dimensions and ${Y}_{p}=1.124\ifmmode\pm\else\textpm\fi{}0.004$ ($\ensuremath{\nu}=0.89$) in three dimensions. This weight function allows for easy analysis of percolation in higher dimensions.
A renormalization group approach in real space is applied to the bond percolating square lattice in two dimension using two scaling transformations proposed by Reynolds et al. (Reynolds's method) and by Yuge (Yuge's method). The renormalization transformation R b ( p ), the eigenvalue λ( b ) and the correlation length critical exponent v ( b ) are calculated exactly by changing the scale of length b =4 for Reynolds's method and b=5/2b=5/\sqrt{2}b=5/2​ and 6/26/\sqrt{2}6/2​ for Yuge's method. The true critical exponent v is estimated v =1.341 and 1.337 by extrapolation, respectively. Discussions are made on the results.
We present a histogram Monte Carlo method to calculate the existence probability Ep{\mathit{E}}_{\mathit{p}}Ep​, the percolation probability P, and the mean-cluster size S for percolation problems as continuous functions of the bond or site probability p. We then use Ep{\mathit{E}}_{\mathit{p}}Ep​ and P in a percolation renormalization-group method to calculate the critical point and exponents. Our method gives quite accurate results for percolation problems.