Bond Percolation

In discrete percolation theory, bond percolation is a percolation model on a regular point lattice  in -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  of edges of  to be the set

(1)

and designate each edge of  to be independently "open" with probability  and closed with probability . Next, define an open path to be any path in  all of whose edges are open, and define the so-called open cluster  to be the connected component of the random subgraph of  consisting of only open edges and containing the vertex . Write . The main objects of study in the bond percolation model are then the percolation probability

(2)

and the critical probability

(3)

where  is defined to be the product measure

(4)

 is the Bernoulli measure which assigns  whenever  is closed and assigns  when  is open, and  is the percolation threshold. Bond models for which  will have infinite connected components (i.e., percolations) whereas those for which  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.

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.