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Acta Physica Sinica

ONE-DIMENSIONAL PERCOLATION PROBLEM WITH FURTHER NEIGHBOUR BONDS——REAL-SPACE RENORMALIZATION GROUP METHOD

Tiecheng Li, Zhao-Qing Zhang, PU FU-CHO, 中国科学院物理研究所

Published Jan 1, 1983• 0 citations

Abstract

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.

DOI: 10.7498/aps.32.1273

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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 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.

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 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.

In complete analogy with thermal critical phenomena, it is expected that anisotropic percolation is in the same universality class as the isotropic one. However the previous result, obtained from the closed form conventional cell real space renormalization group method on the square lattice, the isotropic fixed point is completely unstable is known. We examine this in the large cell limit by using a Monte Carlo renormalization group method and show that the scaling exponent associated with anisotropy is not relevant.

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/\sqrt{2}

and

6/\sqrt{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.