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.

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.