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Physical review. B, Condensed matter

Critical exponents for two-dimensional bond percolation

Christopher J. Lobb, Keith R. Karasek

Published Jan 1, 1982• 13 citations
Publisher

Abstract

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 $\ensuremath{\eta}=0.200\ifmmode\pm\else\textpm\fi{}0.034$.

DOI
10.1103/physrevb.25.492

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Physical review. B, Condensed matter•Aug 1, 1978•17 citations
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 yp=0.750\ifmmode±\else\textpm\fi0.005{y}_{p}=0.750\ifmmode\pm\else\textpm\fi{}0.005yp​=0.750\ifmmode±\else\textpm\fi0.005 and yh=1.90\ifmmode±\else\textpm\fi0.017{y}_{h}=1.90\ifmmode\pm\else\textpm\fi{}0.017yh​=1.90\ifmmode±\else\textpm\fi0.017. This leads, via scaling, to critical exponents \ensuremathα=\ensuremath−0.667\ifmmode±\else\textpm\fi0.018\ensuremath{\alpha}=\ensuremath{-}0.667\ifmmode\pm\else\textpm\fi{}0.018\ensuremathα=\ensuremath−0.667\ifmmode±\else\textpm\fi0.018, \ensuremathβ=0.133\ifmmode±\else\textpm\fi0.023\ensuremath{\beta}=0.133\ifmmode\pm\else\textpm\fi{}0.023\ensuremathβ=0.133\ifmmode±\else\textpm\fi0.023, \ensuremathγ=2.400\ifmmode±\else\textpm\fi0.048\ensuremath{\gamma}=2.400\ifmmode\pm\else\textpm\fi{}0.048\ensuremathγ=2.400\ifmmode±\else\textpm\fi0.048, \ensuremathδ=19.0\ifmmode±\else\textpm\fi3.4\ensuremath{\delta}=19.0\ifmmode\pm\else\textpm\fi{}3.4\ensuremathδ=19.0\ifmmode±\else\textpm\fi3.4, \ensuremathν=1.333\ifmmode±\else\textpm\fi0.009\ensuremath{\nu}=1.333\ifmmode\pm\else\textpm\fi{}0.009\ensuremathν=1.333\ifmmode±\else\textpm\fi0.009, and \ensuremathη=0.200\ifmmode±\else\textpm\fi0.034\ensuremath{\eta}=0.200\ifmmode\pm\else\textpm\fi{}0.034\ensuremathη=0.200\ifmmode±\else\textpm\fi0.034.
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 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 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.
We propose a real-space renormalization-group approach for the bond-percolation problem in a square lattice with first- and second-neighbor bonds. We treat the respective probabilities as independent variables. Two types of cells are constructed. In one of them we consider the lattice as two interpenetrating sublattices, first-neighbor bonds playing the role of intersublattice links. This allows the calculation of both critical exponents $\ensuremath{\nu}$ and $\ensuremath{\gamma}$, without resorting to any external field. Values found for the critical indices are in good agreement with data available in the literature. The phase diagram in parameter space is also obtained in each case.
A renormalization-group approach using a scaling transformation in real space is applied to the critical behavior of two-dimensional percolation systems. In various approximations for the triangular-site lattice and the square-bond lattice, the location of the fixed point and the correlation-length exponent are calculated by determining the behavior of the probabilities under a scale transformation. The fixed points for all approximations for the two lattices are found to be in complete agreement with known exact results for the critical percolation probability.