Large-cell Monte Carlo renormalization group for percolation

Abstract

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 $p_c\mathrm{}\left(b\right)\mathrm{}$ and for the scaling powers $y_p\mathrm{}\left(b\right)\mathrm{}$ and $y_h\mathrm{}\left(b\right)\mathrm{}$. Extrapolating these sequences to the limit $b\to \infty$ 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 $y_p=0.7385\mathrm{}\pm 0.0080$ and $y_h=1.898\mathrm{}\pm 0.003$. Thus we find ${\alpha }_p=-0.708\mathrm{}\pm 0.030$, ${\beta }_p=0.138\mathrm{}(+0.006,-0.005\mathrm{})\mathrm{}$, ${\gamma }_p=2.432\mathrm{}\pm 0.035$, ${\delta }_p=18.6\mathrm{}\pm 0.6$, ${\nu }_p=1.354\mathrm{}\pm 0.015$, and $2-{\eta }_p=1.796\mathrm{}\pm 0.006$. The site-percolation threshold is found for the square lattice at $p_c=0.5931\mathrm{}\pm 0.0006$. We note that our calculated value of ${\nu }_p$ is in considerably better agreement with the proposal of Klein et al. that ${\nu }_p=\frac{\ln \sqrt{3}}{\ln \left(\frac{3}{2}\right)}\cong 1.3548$, than with den Nijs' recent conjecture, which predicts ${\nu }_p=\frac{4}{3}$. However, our results cannot entirely rule out the latter possibility.