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Physical review. B, Condensed matter

Renormalization-group approach for critical percolation behavior in two dimensions

Y Yuge

Published Aug 1, 1978• 17 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.18.1514

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Physical Review Letters•Nov 16, 1981•11 citations
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 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.
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 recently developed renormalization group approach to two-dimensional bond percolation problems is generalized to deal with the percolation conductivity sigma of dilute resistor networks. The critical exponent t, defined by sigma (p) varies as (p-p c ) t , is related to an eigenvalue of the renormalization group transformation of conductance probability distributions. The transformation and eigenvalue are obtained for various real-space scalings of the square and triangular lattices, resulting in t=1.13+or-0.09. The Last-Kirkpatrick scaling relationship is used to obtain the critical exponent for spin-wave stiffness in the dilute ferromagnet.
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 new real space renormalization group transformation combining ideas from mean-field and finite-size scaling theories is presented. Application to the two-dimensional site directed percolation problem gives better values for the percolation threshold and critical exponent of longitudinal correlation length than those obtained previously with other real space renormalization group approaches. When applied to one-, two- and three-dimensional) Ising models, the results are comparable to the ones from previous mean-field-like renormalization group transformations. This method can easily be applied to other systems having second-order phase transitions.
A two-parameter real-space renormalization-group approach to the site-percolation problem in a square lattice is proposed. The two parameters are the concentration ppp of active first-neighbor bonds, and the concentration qqq of occupied sites. From these two parameters one can calculate two independent critical exponents, and the others can be obtained through scaling relations, with no need of external applied fields.