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Physical review. B, Condensed matter

Correlation-length exponent in two-dimensional percolation and Potts model

P. D. Eschbach, Dietrich Stauffer, Hans Jürgen Herrmann

Published Jan 1, 1981• 84 citations
Publisher

Abstract

Following Reynolds et al. and Swendsen, Monte Carlo renormalization calculations were made for percolation on the triangular lattice for rescaling up to 10000\ifmmode\times\else\texttimes\fi{}10000, and for the 4-state Potts model on square lattices up to 64\ifmmode\times\else\texttimes\fi{}64. In contrast to earlier studies, our comparison with the den Nijs hypothesis gives good agreement with its prediction $y=\frac{3}{4}$ for percolation but bad agreement with its prediction $y=\frac{3}{2}$ for the 4-state Potts model.

DOI
10.1103/physrevb.23.422

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Following Reynolds et al. and Swendsen, Monte Carlo renormalization calculations were made for percolation on the triangular lattice for rescaling up to 10000\ifmmode\times\else\texttimes\fi{}10000, and for the 4-state Potts model on square lattices up to 64\ifmmode\times\else\texttimes\fi{}64. In contrast to earlier studies, our comparison with the den Nijs hypothesis gives good agreement with its prediction y=34y=\frac{3}{4}y=43​ for percolation but bad agreement with its prediction y=32y=\frac{3}{2}y=23​ for the 4-state Potts model.
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 two-dimensional Potts models were simulated with the use of a Monte Carlo method. This study is unique in that we simulated the weighted percolation clusters first described by Kasteleyn and Fortuin in 1969. We describe an auxiliary data structure which enables us to determine the connectedness of large clusters very efficiently. Our simulation of the percolation problem was found not to be affected by a critical slowing down. Critical exponents were found by studying the size and shape of large clusters. Our results agree with previous studies done on integer values of qqq, and with the conjecture of den Nijs.
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.