Bond Percolation


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

First, define the set E=E^d of edges of L to be the set

 E={{x,y}:x,y in L^d,|x-y|=1},

and designate each edge of E to be independently "open" with probability p in [0,1] and closed with probability q=1-p. Next, define an open path to be any path in L all of whose edges are open, and define the so-called open cluster C(x) to be the connected component of the random subgraph of L consisting of only open edges and containing the vertex x in L. Write C=C(0). The main objects of study in the bond percolation model are then the percolation probability


and the critical probability


where P_p is defined to be the product measure

 P_p=product_(e in E^d)mu_e,

mu_e is the Bernoulli measure which assigns q=1-p whenever e is closed and assigns p when e is open, and p_c is the percolation threshold. Bond 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, bond percolation is considered less general than site 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 bond percolation; for example, one could drop the assumption of independence to obtain a non-Bernoulli, dependent bond model.

See also

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

This entry contributed by Christopher Stover

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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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Bond Percolation

Cite this as:

Stover, Christopher. "Bond Percolation." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein.

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