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Journal of Physics A Mathematical and General

Renormalisation-group results for bond and site percolation in two and three dimensions

Theodore W. Burkhardt, B. W. Southern

Published Oct 1, 1978• 8 citations
Publisher

Abstract

Critical exponents and probabilities are calculated for bond and site percolation on the d=2 square lattice and the d=3 BCC lattice. The calculation exploits the relationship between the percolation problem and the s-state Potts model in the limit s to 1. The s-state Potts model is analysed using Kadanoff's variational renormalisation transformation. Results for the critical exponents and critical couplings of the s-state Potts model in the interval 0<or=s<or=1 are also presented.

DOI
10.1088/0305-4470/11/10/004

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Journal of the Physical Society of Japan•Apr 1, 1981•0 citations
Critical exponents and probabilities are calculated for bond and site percolation on the d=2 square lattice and the d=3 BCC lattice. The calculation exploits the relationship between the percolation problem and the s-state Potts model in the limit s to 1. The s-state Potts model is analysed using Kadanoff's variational renormalisation transformation. Results for the critical exponents and critical couplings of the s-state Potts model in the interval 0<or=s<or=1 are also presented.
We study the site-bond-correlated percolation model in two and three dimensions. In this model, two first-neighbor sites are considered to be joined by an active bond if both are present and if their first neighbors, along the line joining them, are also present. Using a Monte Carlo renormalization-group approach, we calculate percolation thresholds and critical exponents for both square and simple cubic lattices. Our results indicate that the correlated model is in the same universality class as usual percolation, in two and three dimensions; the critical probabilities for the square and simple cubic lattices are found to be 0.7402\ifmmode\pm\else\textpm\fi{}0.0002 and 0.5660\ifmmode\pm\else\textpm\fi{}0.0004, respectively.
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.
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.
A Monte Carlo renormalisation group approach using a scaling transformation in real space is applied to the critical properties for the site percolation problem on the simple cubic lattice. The authors find a sequence of estimates for the critical concentration p c and the eigenvalue lambda at various values of the rescaling length b. Extrapolation of these sequences to the limit b to infinity yields the site percolation threshold p c =0.3115+0.0004/-0.0003 and the connectedness length exponent nu =0.88+or-0.04.
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.