Random-Connection Model

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 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 models 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 Euclidean l²-norm 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 an -valued point process, with defined below, on a probability space . Next, define the space to be the product

(1)

where the product is taken over all unordered pairs of distinct dyadic half-open hypercubes , with a nonnegative integer and . Equip with the usual product measure under which each coordinate has the marginal distribution given by Lebesgue measure on . 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 dyadic half-open hypercube 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 distinct points the dyadic half-open hypercubes

(3)

and

(4)

respectively, whereby the points and are connected iff

(5)

where denotes the coordinate of the element indexed by the indicated pair of dyadic half-open hypercubes. Using this construction, one gets a collection of vertices in -dimensional Euclidean space which are connected according to the above-described construction.

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. To define the component of the origin, adjoin to if it is not already present and connect the added vertex to each independently with probability , using random choices independent of the existing edges. The component containing in this augmented graph is denoted and is considered to be this model's analog of the notion of occupied components from Boolean models. There is no analog of that model's notion of vacant components.

The above figure illustrates a realization of a random-connection model and some of its terminology. In this figure, the component has the form , the edges of which have the form , , and .