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Physical Review Letters

Site Percolation in a Square Lattice with Second- and Dilute First-Neighbor Bonds

P.M.C. de Oliveira

Published Nov 16, 1981• 11 citations
Publisher

Abstract

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 $p$ of active first-neighbor bonds, and the concentration $q$ 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.

DOI
10.1103/physrevlett.47.1423

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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.
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 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.
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.
One-dimensional site and bond percolation problems with bonds connecting Lth nearest neighbors are studied by using real-space renormalization group method. Exact thermal-like and field-like scaling powers are found. Using the scaling relations we obtain all the critical exponents. For the site percolation, we have ap= 2-L, βp=0,γp= L, δp= ∞, ηp = 1 and vp = L which are consistent with the results obtained by using generating function method. For the bond percolation, we have ap = 2-(L(L+1))/2,βp= 0, γp= L, δp =∞, ηp= 1 and vp= L where the "thermal" exponents are consistent with the results obtained by using transfer matrix method. Magnetic exponents found here are new results. Suzuki's renormalized ex-ponents are φ≡(2-α)/v = 1, β≡β/v = 0, γ≡γ/v = 1, δ≡δ=∞ and η≡η=1 which are independent of both L and site or bond percolations.
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.
We consider the cluster-size statistics of two types of site-bond-correlated percolation. In model A (B) [see Phys. Rev. B 40, 10 986 (1989)], the activation of a bond between two first-neighbor occupied sites is also dependent on the occupancy of both (either) of their own nearest neighbors, along the line joining them. The critical concentration for models A and B, obtained from simulations on square lattices of sizes up to $2000\ifmmode\times\else\texttimes\fi{}2000$, is found to be $0.7405\ifmmode\pm\else\textpm\fi{}0.0005$ and $0.6145\ifmmode\pm\else\textpm\fi{}0.0005$, respectively. The resulting critical exponents indicate that both models are in the same universality class as ordinary independent (site or bond) percolation.