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Physical review. B, Condensed matter

Analytic solution of the block-cluster theory of site and bond percolation in arbitrary dimension

Garnet N. Ord, B. Payandeh, M. Robert

Published Jan 1, 1988• 5 citations
Publisher

Abstract

The percolation transition in arbitrary dimension is studied in real space by generalizing the block-cluster theory. For both site and bond percolation on a decorated hypercubic lattice, this theory yields analytic renormalization-group equations in any number of dimensions. For bond percolation, these analytic equations are most remarkably simple. Results are presented for the thermal exponent and compare very favorably with results obtained by other methods. This is the first successful renormalization-group theory in real space in which the dimension of space enters as a variable parameter.

DOI
10.1103/physrevb.37.467

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The percolation transition in arbitrary dimension is studied in real space by generalizing the block-cluster theory. For both site and bond percolation on a decorated hypercubic lattice, this theory yields analytic renormalization-group equations in any number of dimensions. For bond percolation, these analytic equations are most remarkably simple. Results are presented for the thermal exponent and compare very favorably with results obtained by other methods. This is the first successful renormalization-group theory in real space in which the dimension of space enters as a variable parameter.
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.
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.
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 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.