Bond percolation is a model in discrete percolation theory on a regular point lattice 
in 
-dimensional Euclidean
space that takes 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 ![p in [0,1]](/images/equations/BondPercolation/Inline6.svg)
and closed with probability 
. Next, define an open graph
path to be any graph 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 graph 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 
have infinite connected
components (i.e., percolations) almost
surely, whereas those for which 
have none almost surely.
For nearest-neighbor Bernoulli bond percolation on 
, the critical-value question asks whether
 |
(5)
|
An AI-generated Lean 4 development released by Anthropic (2026) claims this equality for every integer 
, including the previously unresolved dimensions
3 through 10. The formal theorem concerns the value at
; it does not directly state continuity
of 
throughout ![[0,1]](/images/equations/BondPercolation/Inline26.svg)
.
The repository reports kernel checking and adversarial AI review, but no independent
human verification had been reported as of Sep. 9, 2026.
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 graph 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.
