Fast Monte Carlo algorithm for site or bond percolation.

@article{Newman2001FastMC,
  title={Fast Monte Carlo algorithm for site or bond percolation.},
  author={Mark E. J. Newman and Robert M. Ziff},
  journal={Physical review. E, Statistical, nonlinear, and soft matter physics},
  year={2001},
  volume={64 1 Pt 2},
  pages={
          016706
        }
}
  • M. NewmanR. Ziff
  • Published 18 January 2001
  • Physics
  • Physical review. E, Statistical, nonlinear, and soft matter physics
We describe in detail an efficient algorithm for studying site or bond percolation on any lattice. The algorithm can measure an observable quantity in a percolation system for all values of the site or bond occupation probability from zero to one in an amount of time that scales linearly with the size of the system. We demonstrate our algorithm by using it to investigate a number of issues in percolation theory, including the position of the percolation transition for site percolation on the… 
View on PubMed
arxiv.org
374 Citations
Highly Influential Citations
31
Background Citations
55
Methods Citations
99
Results Citations
3

374 Citations

Percolation thresholds for discrete-continuous models with nonuniform probabilities of bond formation.

A class of discrete-continuous percolation models and an efficient Monte Carlo algorithm for computing their properties are introduced and it is found that it compares favorably to well-known algorithms for simpler systems.

Suppressed bond-site percolation

Percolation on two- and three-dimensional lattices.

A highly efficient Monte Carlo algorithm recently proposed by Newman and Ziff is applied to treat percolation problems to confirm the universal aspect of the wrapping probabilities regarding site and bond dilution.

Percolation of the site random-cluster model by Monte Carlo method.

A site random-cluster model is proposed by introducing an additional cluster weight in the partition function of the traditional site percolation by combining the color-assignation and the Swendsen-Wang methods to design a highly efficient cluster algorithm with a small critical slowing-down phenomenon.

Convergence of threshold estimates for two-dimensional percolation.

  • R. ZiffM. Newman
  • Mathematics
    Physical review. E, Statistical, nonlinear, and soft matter physics
  • 2002
This work shows that the convergence of the average-probability estimate is described by a nontrivial correction-to-scaling exponent as predicted previously, and measures the value of this exponent to be 0.90+/-0.02.

Dimer site-bond percolation on a triangular lattice

A generalization of the site-percolation problem, in which pairs of neighbor sites (site dimers) and bonds are independently and randomly occupied on a triangular lattice, has been studied by means…

Site-bond percolation on simple cubic lattices: numerical simulation and analytical approach

The site-percolation problem on simple cubic lattices has been studied by means of numerical simulation and analytical calculations based on exact counting of configurations on finite cells.…

Site percolation on lattices with low average coordination numbers

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…

Computational studies of bond-site percolation.

Percolation theory enters in various areas of research including critical phenomena and phase transitions. Bond-site percolation is a generalization of pure percolation motivated by the fact that…

Bond percolation on simple cubic lattices with extended neighborhoods.

The results show that the percolation thresholds of these and other three-dimensional lattices decrease monotonically with the coordination number z quite accurately according to a power-law p_{c}∼z^{-a} with exponent a=1.111.
...

References

SHOWING 1-10 OF 59 REFERENCES

Histogram Monte Carlo renormalization-group method for percolation problems.

  • Hu
  • Physics
    Physical review. B, Condensed matter
  • 1992
We present a histogram Monte Carlo method to calculate the existence probability E p , the percolation probability P, and the mean-cluster size S for percolation problems as continuous functions of…

PERCOLATION AS A DYNAMICAL PHENOMENON

Exact solution of site and bond percolation on small-world networks.

  • C. MooreM. Newman
  • Mathematics
    Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics
  • 2000
An exact solution of the model for both site and bond percolation is given, including the position of thePercolation transition at which epidemic behavior sets in, the values of the critical exponents governing this transition, the mean and variance of the distribution of cluster sizes below the transition, and the size of the giant component (epidemic) above the transition.

Number of bonds in the site-diluted lattices: sampling and fluctuations

We have calculated analytically the mean value and the variance of the number of bonds on the lattices of dimension $d$ for the given occupation of sites. We consider both kinds of site occupation:…

Properties of percolation clusters in a model granular system in two dimensions

Properties of percolation clusters in a model granular system on a square lattice are examined by a Monte Carlo simulation. Grains are grown around seeds up to the nth neighbours. The critical…

Percolation by position-space renormalisation group with large cells

We demonstrate that the critical properties of the percolation model can be obtained with a rather high degree of accuracy by extrapolating to b = a3 a sequence of position-space renormalisation…

Generation of percolation cluster perimeters by a random walk

A type of self-avoiding random walk whish generates the perimeter of two- dimensional lattice-percolation clusters is given. The algorithm has been simulated on a computer, yielding the mean…

Scaling and universality in the spanning probability for percolation.

  • HoviAharony
  • Physics
    Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics
  • 1996
This paper discusses the spanning percolation probability function for three different spanning rules, in general dimensions, with both free and periodic boundary conditions, and finds strong relations among different derivatives of the spanning function with respect to the scaling variables, thus yielding several universal amplitude ratios.

Scaling and percolation in the small-world network model.

  • M. NewmanD. Watts
  • Mathematics
    Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics
  • 1999
There is one nontrivial length-scale in the small-world network model of Watts and Strogatz, analogous to the correlation length in other systems, which is well-defined in the limit of infinite system size and which diverges continuously as the randomness in the network tends to zero, giving a normal critical point in this limit.

Universality of the excess number of clusters and the crossing probability function in three-dimensional percolation

Extensive Monte Carlo simulations were performed to evaluate the excess number of clusters and the crossing probability function for three-dimensional percolation on the simple cubic (s.c.),…
...