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Site Percolation


SitePercolation

In discrete percolation theory, site percolation is a percolation model on a regular point lattice L=L^d in d-dimensional Euclidean space which considers the lattice vertices as the relevant entities (left figure). The precise mathematical construction for the Bernoulli version of site percolation is as follows.

First, designate each vertex of L to be independently "open" with probability p in [0,1] and closed otherwise. Next, define an open path to be any path in L all of whose vertices are open, and define at the vertex x in L the so-called open cluster C(x) to be the set of all vertices which may be attained following only open paths from x. Write C=C(0). The main objects of study in the site percolation model are then the percolation probability

 theta(p)=P_p(|C|=infty)
(1)

and the critical probability

 p_c=sup{p:theta(p)=0}
(2)

where here, P_p is defined to be the product measure

 P_p=product_(v in L^d)mu_v,
(3)

mu_v is the Bernoulli measure which assigns q=1-p whenever v is closed and assigns p when v is open, and p_c is the percolation threshold. Site models for which p>p_c will have infinite connected components (i.e., percolations) whereas those for which p<p_c will not.

In general, site percolation is considered more general than bond percolation due to the fact that every bond model may be reformulated as a site model on a different lattice but not vice versa. Mixed percolation is considered to be a bridge between the two. Note, too, the existence of several other variants of site percolation; for example, one could drop the assumption of independence to obtain a non-Bernoulli, dependent site model.


See also

Dependent Percolation, Bond Percolation, Discrete Percolation Theory, Mixed Percolation Model, Percolation, Percolation Theory, Percolation Threshold

Portions of this entry contributed by Christopher Stover

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References

Chayes, L. and Schonmann, R. H. "Mixed Percolation as a Bridge Between Site and Bond Percolation." Ann. Appl. Probab. 10, 1182-1196, 2000.Grimmett, G. Percolation, 2nd ed. Berlin: Springer-Verlag, 1999.Hammersley, J. M. "A Generalization of McDiarmid's Theorem for Mixed Bernoulli Percolation." Math. Proc. Camb. Phil. Soc. 88, 167-170, 1980.

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Site Percolation

Cite this as:

Stover, Christopher and Weisstein, Eric W. "Site Percolation." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/SitePercolation.html

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