Site-bond percolation problems

We develop a real-space renormalization group 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. Our results are in excellent agreement with series calculations for the critical percolation concentration p, (both site and bond), and in good agreement for the correlation length exponent vp. Additionally, in one dimension we include a field-like variable and calculate the remaining exponents.

A generalisation of the pure site and pure bond percolation problems is studied, in which both the sites and bonds are independently occupied at random. This generalisation-the site-bond problem-is of current interest because of its application to the phenomenon of polymer gelation. Motivated by considerations of cluster connectivity, we have defined two distinct models for site-bond percolation, models A and B. In model A, a cluster is considered to be a set of occupied bonds and sites in which the bonds are joined by occupied sites, and the sites are joined by occupied bonds. Since a bond cannot contribute to cluster connectivity if either site at its endpoints is not occupied, we define model B in which these 'non-connecting' bonds are treated as part of the cluster perimeter. We prove that the critical curve and critical exponents are the same for both models. For model B, we calculate low-density series expansions for the mean cluster size on the square lattice. We calculate three different series, using the following definitions of cluster size: site size, bond size, and a hybrid measure involving both site and bond size. All three series have been used to obtain the phase boundary between the percolating (gel) and non-percolating (sol) regions. Numerical evidence is presented which indicates that along the entire phase boundary the mean-size exponent y assumes a universal value.

Physical review. E, Statistical, nonlinear, and soft matter physics, 2003

In this work we apply a highly efficient Monte Carlo algorithm recently proposed by Newman and Ziff to treat percolation problems. The site and bond percolations are studied on a number of lattices in two and three dimensions. Quite good results for the wrapping probabilities, correlation length critical exponent, and critical concentration are obtained for the square, simple cubic, hexagonal close packed, and hexagonal lattices by using relatively small systems. We also confirm the universal aspect of the wrapping probabilities regarding site and bond dilution.

Physics Letters A, 2003

The effects of the dilution of sites (with an occupancy probability p s for an active site) on the long-range bond-percolation problem, on a linear chain (d = 1), are analyzed by means of a Monte Carlo simulation. The occupancy probability for a bond between two active sites i and j , separated by a distance r ij is given by p ij = p/r α ij (α 0), where p represents the usual occupancy probability between nearest-neighbor sites. The percolation order parameter, P ∞ , is investigated numerically for different values of α and p s , in such a way that a crossover between a nonextensive regime and an extensive regime is observed.

2001

Percolation models with multiple percolating clusters have attracted much attention in recent years. Here we use Monte Carlo simulations to study bond percolation on L_1× L_2 planar random lattices, duals of random lattices, and square lattices with free and periodic boundary conditions, in vertical and horizontal directions, respectively, and with various aspect ratio L_1/L_2. We calculate the probability for the appearance of n percolating clusters, W_n, the percolating probabilities, P, the average fraction of lattice bonds (sites) in the percolating clusters, <c^b>_n (<c^s>_n), and the probability distribution function for the fraction c of lattice bonds (sites), in percolating clusters of subgraphs with n percolating clusters, f_n(c^b) (f_n(c^s)). Using a small number of nonuniversal metric factors, we find that W_n, P, <c^b>_n (<c^s>_n), and f_n(c^b) (f_n(c^s)) for random lattices, duals of random lattices, and square lattices have the same universal fi...

Physical Review B, 1980

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The Annals of Applied Probability, 2000

By using mixed percolation as a bridge between site and bond percolation, we derive a new inequality between the critical points of these processes that is optimal in a certain sense. We also extend a result on the crossover exponent of bond-diluted Potts models to site-diluted Potts models. Some new results about the critical line in mixed percolation are also proved.

Journal of Statistical Mechanics: Theory and Experiment, 2014

We present a study of site and bond percolation on periodic lattices with (on average) fewer than three nearest neighbors per site. We have studied this issue in two contexts: By simulating oxides with a mixture of 2-coordinated and higher-coordinated sites, and by mapping site-bond percolation results onto a site model with mixed coordination number. Our results show that a conjectured power-law relationship between coordination number and site percolation threshold holds approximately if the coordination number is defined as the average number of connections available between high-coordinated sites, and suggest that the conjectured power-law relationship reflects a real phenomenon requiring further study. The solution may be to modify the power-law relationship to be an implicit formula for percolation threshold, one that takes into account aspects of the lattice beyond spatial dimension and average coordination number.

Physica Scripta, 2011

For certain hierarchical structures, one can study the percolation problem using the renormalization-group method in a very precise way. We show that the idea can be also applied to two-dimensional planar lattices by regarding them as hierarchical structures. Either a lower bound or an exact critical probability can be obtained with this method and the correlation-length critical exponent is approximately estimated as ν ≈ 1.

Physical Review B, 1984

A method for generalizing bond-percolation problems to include the possibility of infinite-range (equivalent-neighbor) bonds is presented. On Bravais lattices the crossover from nonclassical to classical (mean-field) percolation criticality in the presence of such bonds is described. The Cayley tree with nearest-neighbor and equivalent-neighbor bonds is solved exactly, and a nonuniversal line of percolation transitions with exponents dependent on nearest-neighbor bond occupation probability is observed. Points of logarithmic and exponential singularity are also encountered, and the behavior is interpreted as dimensional reduction due to the breaking of translational invariance by bonds of Cayley-tree connectivity.