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
![E=<span style=]() {{x,y}:x,y in L^d,|x-y|=1}, " src="https://mathworld.wolfram.com/images/equations/BondPercolation/NumberedEquation1.svg" style="height:25px; width:240px" /> |
(1)
|
and designate each edge of 
to be independently "open" with probability ![p in [0,1]](https://mathworld.wolfram.com/images/equations/BondPercolation/Inline6.svg)
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
![p_c=sup<span style=]() {p:theta(p)=0}, " src="https://mathworld.wolfram.com/images/equations/BondPercolation/NumberedEquation3.svg" style="height:21px; width:159px" /> |
(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.
