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

Physical review. B, Condensed matter

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

Naeem Jan, D. L. Hunter, Turab Lookman

Published Jun 1, 1984• 6 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.29.6356

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

Christopher J. Lobb, Keith R. Karasek

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.

Journal of Physics C Solid State Physics•Apr 20, 1980•13 citations

Large-cell Monte Carlo renormalization group for percolation

PETER J. REYNOLDS, Eugene Stanley, W. Klein

We obtain the critical parameters for the site-percolation problem on the square lattice to a high degree of accuracy (comparable to that of series expansions) by using a Monte Carlo position-space renormalization-group procedure directly on the site-occupation probability. Our method involves calculating recursion relations using progressively larger lattice rescalings, $b$. We find smooth sequences for the value of the critical percolation concentration ${p}_{c}(b)$ and for the scaling powers ${y}_{p}(b)$ and ${y}_{h}(b)$. Extrapolating these sequences to the limit $b\ensuremath{\rightarrow}\ensuremath{\infty}$ leads to quite accurate numerical predictions. Further, by considering other weight functions or "rules" which also embody the essential connectivity feature of percolation, we find that the numerical results in the infinite-cell limit are in fact "rule independent." However, the actual fashion in which this limit is approached does depend upon the rule chosen. A connection between extrapolation of our renormalization-group results and finite-size scaling is made. Furthermore, the usual finite-size scaling arguments lead to independent estimates of ${p}_{c}$ and ${y}_{p}$. Combining both the large-cell approach and the finite-size scaling results, we obtain ${y}_{p}=0.7385\ifmmode\pm\else\textpm\fi{}0.0080$ and ${y}_{h}=1.898\ifmmode\pm\else\textpm\fi{}0.003$. Thus we find ${\ensuremath{\alpha}}_{p}=\ensuremath{-}0.708\ifmmode\pm\else\textpm\fi{}0.030$, ${\ensuremath{\beta}}_{p}=0.138(+0.006,\ensuremath{-}0.005)$, ${\ensuremath{\gamma}}_{p}=2.432\ifmmode\pm\else\textpm\fi{}0.035$, ${\ensuremath{\delta}}_{p}=18.6\ifmmode\pm\else\textpm\fi{}0.6$, ${\ensuremath{\nu}}_{p}=1.354\ifmmode\pm\else\textpm\fi{}0.015$, and $2\ensuremath{-}{\ensuremath{\eta}}_{p}=1.796\ifmmode\pm\else\textpm\fi{}0.006$. The site-percolation threshold is found for the square lattice at ${p}_{c}=0.5931\ifmmode\pm\else\textpm\fi{}0.0006$. We note that our calculated value of ${\ensuremath{\nu}}_{p}$ is in considerably better agreement with the proposal of Klein et al. that ${\ensuremath{\nu}}_{p}=\frac{\mathrm{ln}\sqrt{3}}{\mathrm{ln}(\frac{3}{2})}\ensuremath{\cong}1.3548$, than with den Nijs' recent conjecture, which predicts ${\ensuremath{\nu}}_{p}=\frac{4}{3}$. However, our results cannot entirely rule out the latter possibility.

Physical review. B, Condensed matter•Feb 1, 1980•457 citations

Fragmentation scaling of percolation clusters in two and three dimensions: Large-cell Monte Carlo RG approach

Mookyung Cheon, Iksoo Chang

The scaling behavior for a binary fragmentation of critical percolation clusters is investigated by a large-cell Monte Carlo real-space renormalization group method in two and three dimensions. We obtain accurate values of critical exponents λ and ϕ describing the scaling of fragmentation rate and the distribution of fragments' masses produced by a binary fragmentation. Our results for λ and ϕ show that the fragmentation rate is proportional to the size of mother cluster, and the scaling relation σ = 1 + λ − ϕ conjectured by Edwards et al. to be valid for all dimensions is satisfied in two and three dimensions, where σ is the crossover exponent of the average cluster number in percolation theory, which excludes the other scaling relations.

Europhysics Letters (EPL)•Apr 1, 1999•2 citations

Monte Carlo renormalization-group analysis of percolation

Albert Brown, Alexander Edelman, Jason W. Rocks +2 more

We describe a Monte Carlo renormalization group approach to the calculation of critical behavior for percolation models. This approach can be utilized to determine the renormalized bond probabilities and the values of the critical exponents. We illustrate the method for two-dimensional bond percolation, but the method is also applicable to other percolation models and other dimensions.

Physical Review E•Oct 22, 2013•4 citations

Evidence for classical critical behavior in long-range site percolation

M.A. Gouker, Fereydoon Family

A new large-cell Monte Carlo renormalization group following the approach of Reynolds et al. is developed to study site percolation with long-range interaction. The results show that site percolation with a long, but finite, range of interactions has the same correlation-length exponent ${\ensuremath{\nu}}_{P}$ as ordinary percolation. However, as the interaction range tends to infinity, ${\ensuremath{\nu}}_{P}$ approaches its mean-field value of $\frac{1}{2}$ indicating that infinite-range percolation exhibits classical critical behavior.

Physical review. B, Condensed matter•Aug 1, 1983•39 citations
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 Yp=0.73\ifmmode±\else\textpm\fi0.01{Y}_{p}=0.73\ifmmode\pm\else\textpm\fi{}0.01Yp​=0.73\ifmmode±\else\textpm\fi0.01 (the correlation length exponent \ensuremathν=1.37\ensuremath{\nu}=1.37\ensuremathν=1.37) in two dimensions and Yp=1.124\ifmmode±\else\textpm\fi0.004{Y}_{p}=1.124\ifmmode\pm\else\textpm\fi{}0.004Yp​=1.124\ifmmode±\else\textpm\fi0.004 (\ensuremathν=0.89\ensuremath{\nu}=0.89\ensuremathν=0.89) in three dimensions. This weight function allows for easy analysis of percolation in higher dimensions.
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 $\ensuremath{\eta}=0.200\ifmmode\pm\else\textpm\fi{}0.034$.
We obtain the critical parameters for the site-percolation problem on the square lattice to a high degree of accuracy (comparable to that of series expansions) by using a Monte Carlo position-space renormalization-group procedure directly on the site-occupation probability. Our method involves calculating recursion relations using progressively larger lattice rescalings, bbb. We find smooth sequences for the value of the critical percolation concentration pc(b){p}_{c}(b)pc​(b) and for the scaling powers yp(b){y}_{p}(b)yp​(b) and yh(b){y}_{h}(b)yh​(b). Extrapolating these sequences to the limit $b\ensuremath{\rightarrow}\ensuremath{\infty}$ leads to quite accurate numerical predictions. Further, by considering other weight functions or "rules" which also embody the essential connectivity feature of percolation, we find that the numerical results in the infinite-cell limit are in fact "rule independent." However, the actual fashion in which this limit is approached does depend upon the rule chosen. A connection between extrapolation of our renormalization-group results and finite-size scaling is made. Furthermore, the usual finite-size scaling arguments lead to independent estimates of pc{p}_{c}pc​ and yp{y}_{p}yp​. Combining both the large-cell approach and the finite-size scaling results, we obtain ${y}_{p}=0.7385\ifmmode\pm\else\textpm\fi{}0.0080$ and ${y}_{h}=1.898\ifmmode\pm\else\textpm\fi{}0.003$. Thus we find ${\ensuremath{\alpha}}_{p}=\ensuremath{-}0.708\ifmmode\pm\else\textpm\fi{}0.030$, ${\ensuremath{\beta}}_{p}=0.138(+0.006,\ensuremath{-}0.005)$, ${\ensuremath{\gamma}}_{p}=2.432\ifmmode\pm\else\textpm\fi{}0.035$, ${\ensuremath{\delta}}_{p}=18.6\ifmmode\pm\else\textpm\fi{}0.6$, ${\ensuremath{\nu}}_{p}=1.354\ifmmode\pm\else\textpm\fi{}0.015$, and $2\ensuremath{-}{\ensuremath{\eta}}_{p}=1.796\ifmmode\pm\else\textpm\fi{}0.006$. The site-percolation threshold is found for the square lattice at ${p}_{c}=0.5931\ifmmode\pm\else\textpm\fi{}0.0006$. We note that our calculated value of ${\ensuremath{\nu}}_{p}$ is in considerably better agreement with the proposal of Klein et al. that ${\ensuremath{\nu}}_{p}=\frac{\mathrm{ln}\sqrt{3}}{\mathrm{ln}(\frac{3}{2})}\ensuremath{\cong}1.3548$, than with den Nijs' recent conjecture, which predicts ${\ensuremath{\nu}}_{p}=\frac{4}{3}$. However, our results cannot entirely rule out the latter possibility.
The scaling behavior for a binary fragmentation of critical percolation clusters is investigated by a large-cell Monte Carlo real-space renormalization group method in two and three dimensions. We obtain accurate values of critical exponents λ and ϕ describing the scaling of fragmentation rate and the distribution of fragments' masses produced by a binary fragmentation. Our results for λ and ϕ show that the fragmentation rate is proportional to the size of mother cluster, and the scaling relation σ = 1 + λ − ϕ conjectured by Edwards et al. to be valid for all dimensions is satisfied in two and three dimensions, where σ is the crossover exponent of the average cluster number in percolation theory, which excludes the other scaling relations.
A new large-cell Monte Carlo renormalization group following the approach of Reynolds et al. is developed to study site percolation with long-range interaction. The results show that site percolation with a long, but finite, range of interactions has the same correlation-length exponent ${\ensuremath{\nu}}_{P}$ as ordinary percolation. However, as the interaction range tends to infinity, ${\ensuremath{\nu}}_{P}$ approaches its mean-field value of 12\frac{1}{2}21​ indicating that infinite-range percolation exhibits classical critical behavior.