A new Monte Carlo method for percolation problems on a lattice

@article{Dean1963ANM,
  title={A new Monte Carlo method for percolation problems on a lattice},
  author={P. Dean},
  journal={Mathematical Proceedings of the Cambridge Philosophical Society},
  year={1963},
  volume={59},
  pages={397 - 410}
}
  • P. Dean
  • Published 1 April 1963
  • Physics
  • Mathematical Proceedings of the Cambridge Philosophical Society
Abstract A new and general Monte Carlo technique is described for solving some well-known percolation and cluster-size problems on regular lattice networks. The method has been applied to ten two-dimensional structures. 

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    Mathematical Proceedings of the Cambridge Philosophical Society
  • 1972
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  • J. Wierman
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The proof of these bounds uses the substitution method, comparing the percolative behaviour of the Kagomé lattice bond model with that of the exactly solved hexagonal lattices bond model via stochastic ordering.
...

References

SHOWING 1-9 OF 9 REFERENCES

Comparison of Atom and Bond Percolation Processes

Various inequalities, some of them strict, are proved concerning probabilities associated with percolation processes. In particular, it is shown that the critical probability of an atom percolation

Critical Probabilities for Cluster Size and Percolation Problems

When particles occupy the sites or bonds of a lattice at random with probability p, there is a critical probability pc above which an infinite connected cluster of particles forms. Rigorous bounds

Some Cluster Size and Percolation Problems

The problem of cluster size distribution and percolation on a regular lattice or graph of bonds and sites is reviewed and its applications to dilute ferromagnetism, polymer gelation, etc., briefly

Remarks on Magnetically Dilute Systems

Because of the inadequacies of previous treatments of the magnetic properties of magnetically dilute systems when used to interpret experimental results, a re‐examination of the problem starting from

On the magnetically dilute Heisenberg and Ising ferromagnetics

The problem of a randomly dilute Heisenberg or Ising ferromagnetic is discussed on the basis of expansions of susceptibility in inverse powers of temperature. The first six significant coefficients

Percolation processes

ABSTRACT The paper studies, in a general way, how the random properties of a ‘medium’ influence the percolation of a ‘fluid’ through it. The treatment diifers from conventional diffusion theory, in