A random-connection model (RCM) is a graph-theoretic model of continuum percolation theory
characterized by the existence of a stationary
point process 
and a non-increasing
function ![g:R^+->[0,1]](/images/equations/Random-ConnectionModel/Inline2.svg)
which together determine a methodology for drawing edges
between various vertex points in 
for some 
.
In this case, the function 
is said to be a connection function and the RCM is said to
be driven by 
. The model itself is denoted by 
.
A generalization of the ordinary nearest-neighbor bond percolation of 
, the random-connection model of continuum percolation should
be regarded as a contrasting alternative to the Boolean
and Boolean-Poisson models, two models in
which the second condition above is replaced with a random
variable 
.
More precisely, the Boolean and Boolean-Poisson models demonstrate percolation
by way of constructing 
-dimensional closed balls centered at points of 
and assigning random radii determined
by 
;
contrarily, an RCM uses notions from graph theory, viewing points from 
as vertices and inserting edges between pairs 
with probability 
,
independently of all other pairs of points
in 
,
where 
denotes the typical Euclidean distance in 
.
Despite their differences, however, the two methods are similar in that sophisticated
machinery is necessary for a mathematically precise presentation of either; the formal
construction for an RCM is as follows.
Let 
be a point process defined on a probability
space 
.
Next, define the space 
to be the product
![Omega_2=product_(K(n,z),K(m,z^'))[0,1]](/images/equations/Random-ConnectionModel/NumberedEquation1.svg) |
(1)
|
where the product is taken over all unordered pairs of binary cubes and define associated to 
the usual product measure 
so that all the marginal probabilities are
given by Lebesgue measure on ![[0,1]](/images/equations/Random-ConnectionModel/Inline24.svg)
. Finally, define 
and equip 
with the product measure 
. Under this construction, an RCM is a measurable
mapping from 
into 
defined by
 |
(2)
|
where here, 
denotes the set of all counting measures on the sigma-algebra 
of Borel sets in 
which assign finite measure to all bounded Borel sets and
which assign values of at most 1 to points.
One then transitions to percolation by first noting that each point 
is contained in a unique binary cube 
of order 
and that, for each 
, there is a unique smallest number 
such that 
contains no other points of 
-almost surely. With this
in mind, consider for any two points 
the binary cubes
 |
(3)
|
and
 |
(4)
|
respectively, whereby the points 
and 
are connected if and only if
 |
(5)
|
where 
is the notation used to denote an element 
. Using this construction, one gets a collection
of vertices in 
-dimensional
Euclidean space which are connected according to a the above-described random process.
Some terminology utilized in random-connection models: The edge between two points 
and 
is denoted by the unordered pair 
and 
are called end vertices of 
. Two points 
are said to be connected if there exists a finite sequence
 |
(6)
|
such that the edge 
is inserted for all 
, and in the typical graph-theoretic way, one then
defines a component as a set 
of points which is maximal with respect to the property that
any two points 
are connected to one another. The occupied component of the origin is denoted 
and is considered to be this model's analogue of the notion of occupied components
from Boolean models. There is no analogue of that model's notion of vacant components.
The above figure illustrates a realization of a random-connection model, illustrating some of the terminology related to thereto. In this figure, the component 
has the form 
, the edges of which have the form 
, 
, and 
.
