Site Percolation Problem Research Articles

Further comments on the one-dimensional percolation problem with multineighbour bonds

One-dimensional site and bond percolation problems with bonds connecting Lth nearest neighbours are solved exactly by using an infinite cell to cell renormalisation group transformation. Both 'thermal' and magnetic critical exponents are obtained through the scaling relations. Although universality breaks down for site and bond percolation when L>or=2, Suzuki's weaker universality (1974) still holds.

One-dimensional percolation problems with further neighbour bonds-transfer-matrix approach

The transfer-matrix method is used to find the exact expressions for the correlation length in the critical region for both one-dimensional site and bond percolation problems with bonds connecting Lth-nearest neighbours (for any finite L). For the site percolation, the correlation length exponent v is found to be L, consistent with the result obtained from the generating function method. For the bond percolation, we find v=L(L+1)/2.

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

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.

Percolation of unfrustrated plaquettes

The author has investigated percolation properties of unfrustrated plaquettes in the two-dimensional Ising model with nearest-neighbour bonds +or-J distributed at random. The effects of plaquette correlations are considered using a two-parameter position-space renormalisation group (PSRG) approach, and it is found (within the framework of this approximate PSRG) that this correlated site percolation problem is in the same universality class as random site percolation. The percolation threshold is then found from large-cell PSRG methods to be indistinguishable from that for randomly placed sites.

Two-dimensional site percolation and conduction

The authors introduce the finite-element method for the effective conductivity of two-dimensional composite materials. For the problem of two-dimensional percolation and conduction they use three kinds of composite material with regular cell structure and compute an upper and a lower bound on the effective conductivity of the materials by the finite-element method. The three models can be related to site percolation problems, and the relation between the computed bounds and two kinds of percolation problems (current percolation and field percolation) is made clear.

Transfer-matrix approach to the one-dimensional percolation problem

The transfer-matrix method is used to find the exact phase diagrams and the correlation length exponents v for the one-dimensional site, bond and site-bond percolation problems with bonds connecting the Lth-nearest neighbours (L up to 3). For the site percolation, the authors' results agree with the exact result obtained from the generating function method, v=L, while for the bond percolation they found much richer critical phenomena. If all the L bonds have equal occupation probability, their results predict v=L(L+1)/2.

General position-space renormalisation group for correlated percolation

A position-space renormalisation group (PSRG) approach has been developed for general site percolation problems in which the site occupancies are correlated, rather than independent, random variables. PSRG parameters are systematically introduced to describe multiple-site correlations. At the two- and four-site levels of approximation in the square lattice the authors find only two physical fixed points. One characterises random, or pure, percolation and has a universality class which is shown to encompass a variety of locally correlated site problems, including unfrustrated plaquettes and four-coordinated sites in a random-bond model. The second describes 'Ising-correlated' percolation and at the two-site level yields an excellent estimate (to within 2%) for the nearest-neighbour spin correlation function of the Ising model.

Combined site and bond percolation on a cayley tree

The combined Ising site and bond percolation problem, i.e., the calculation of the probability of finding an infinite cluster of “up” or “down” spins which are connected by bonds (single bond probability p), is solved exactly for ferromagnetic and antiferromagnetic nearest-neighbour interactions on a Caylee tree. Different types of percolation are shown in p- T phase diagrams.

Site percolation problems and multisite Potts models

The correspondence discovered by Kunz and Wu (1978) between the site percolation problem and the multi-site Potts problem is extended to several lattices. It is found that the site percolation problem can be very simply formulated in terms of the operators used by Temperley and Lieb (1971). In three instances applying the dual transformation to these operators induces the matching transformation found by Sykes and Essam (1964) and it is conjectured that this may hold generally. It follows that site percolation problems, like bond percolation problems, depend on finding eigenvalues of row transfer operators.

Effect of dimensionality on site valence in percolation problems: a comparison of the triangular and simple cubic lattices

The authors present Monte Carlo and series analysis results for the distribution of valences of sites in percolating and non-percolating clusters for the triangular and simple cubic site percolation problems in order to explore the importance of dimensionality, at fixed lattice coordination number, on the degree of ramification of clusters.

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

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.

Temperature-Dependent Percolation in the Ordered Phase–Square Lattice-Gas Model–

Critical percolation density for temperature-dependent site percolation problem is calculated by using the lattice-gas model of atoms, especially inside the region where the sublattice ordering occurs (repulsive interaction) or liquid and gas phases coexist (attractive interaction). The square lattice is regarded as a resistor network. Any bond in the lattice is conductive or insulating whether both ends of the bond are occupied with atoms or not. The distribution of atoms is determined within the pair approximation from the lattice-gas Hamiltonian with nearest-neighbor interactions. Moreover a single-site effective-medium approximation which is similar to coherent potential approximation is used to obtain the critical percolation density.

Diffusion, correlation, and percolation in a random alloy

Abstract The relationship between physical correlation and tracer correlation in a random alloy and their bearing on the site percolation problem are discussed. Four Monte Carlo schemes are described for the calculation of the tracer and physical (vacancy) correlation factors. Results for the latter in the f.c.c. lattice are shown to be in excellent agreement with Manning's theory for concentrations greater than 0.4 of the mobile component. Below that concentration there is a slight deviation as the simulated and predicted vacancy correlation factors tend to zero at slightly different percolation thresholds. The site percolation threshold is determined to be 0·198 ± 0·004, in excellent agreement with previous Monte Carlo studies.

Pure Site Percolation Problem for Linear Ising Chain in the Molecular Field

Pure Site Percolation Problem for Linear Ising Chain in the Molecular Field

A new RG approach to site percolation in two dimensions

A renormalisation group transformation is developed for two-dimensional site-percolation problems by using a scaling transformation in real space. A transition matrix A is defined for each cell, and the renormalised probability p'(p) of occupation of the cell is identified with the dominant eigenvalue R1(p) of the transition matrix. A simple RG transformation has been applied on the square lattice up to cells of size 6*6 and the results for critical probability pc and exponent v are given. A modified RG has been applied to the three planar lattices, and pc and v have been calculated for the simplest choice of the cells. The modified RG transformation seems to yield better results as the coordination number of the lattice increases.

Critical surface for a three-colour site percolation problem on the triangular lattice

A certain relationship between a recent conjecture by F.Y. Wu (1979) and one by W. Klein et al. (1978) is established through considering a three-colour site percolation problem, where the three sublattices of a triangular lattice are populated with probability s1, s2 and s3, respectively. Both conjectures imply the same critical condition for various special cases of this three-colour model, including the threshold probability 1/ square root 2 for the honeycomb site problem. Existing numerical estimates provide strong evidence against this threshold value, hence against both conjectures.

Renormalization-Group Transformation for Site Percolation Problem with Distant-Neighbor Interactions

Effects of distant-neighbor interactions in the site percolation problem are studied in terms of the renormalization-group transformation in square, triangular, simple cubic, body-centered cubic and face-centred cubic lattices. For the square lattice we consider up to the second-, fifth- and eighth-neighbor interactions for the scale factor of the transformations b=2, 3 and 4, respectively. For other lattices the first-, second- and third-neighbor interactions are taken into account. The results for critical probabilities pc, although not for correlation length exponents νp, agree fairly well with the known data. For the square lattice with the first-neighbor interactions and with the first- and second-neighbor interactions, we obtain νp=1.635, 1.625 and 1.607 for b=2, 3 and 4, respectively, which excellently agree with each other for each value of b. In addition, νp for the simple cubic lattice with up to the third-neighbor interactions coincide closely, within 0.6% precision, with each other.

An infinitesimal renormalisation-group transformation for the site percolation problem

The site percolation problem is studied by an infinitesimal (i.e. the scaling factor b to 1) renormalisation transformation. The correlation length exponent v, the 'magnetic' scaling power yh and the three independent conductivity exponents t, s and v are calculated. It is shown that only the shortest one-dimensional percolation paths are important in the derivation of renormalised probabilities and conductances. This explains why two different transformations-one for the site percolation problem and the other for the bond percolation problem-become equivalent in the b to 1 limit.

A Monte Carlo calculation of the cluster size critical exponent for 2D bond percolation

Reynolds, Stanley and Klein (1978) developed a large-cell renormalisation group scheme and applied it to the site percolation problem in two dimensions. Applying this technique to bond percolation, the authors calculate the cluster size critical exponent nu . This method converges very rapidly when applied to the bond problem. They estimate that nu =1.343+or-0.017, in excellent agreement with the earlier site percolation result, thus supporting the idea of site-bond universality.

Correlated percolation on a quadratic lattice

Site percolation on the quadratic lattice is studied in the presence of nearest-neighbour correlations. Firstly, in the presence of strong negative correlations the critical percolation probability is determined by relating it to an uncorrelated site percolation problem, with the result p c(−∞) = 0.705 ± 0.005. Secondly, the dependence of the critical probability upon the interaction is determined by a real-space renormalization method.