Boolean Model

In most modern literature, a Boolean model is a probabilistic model of continuum percolation theory characterized by the existence of a stationary point process X and a random variable rho which independently determine the centers and the random radii of a collection of closed balls in R^d for some d.

In this case, the model is said to be driven by X.

Worth noting is that the most intuitive ideas about constructing a feasible model using X and rho often lead to unexpected and undesirable results (Meester and Roy 1996). For that reason, some more sophisticated machinery and quite a bit of care is needed to translate from the language of X and rho into a reasonable model of continuum percolation. The formal construction is as follows.

Let X be a stationary point process as discussed above and suppose that X is defined on a probability space (Omega_1,F_1,P_1). Next, define the space Omega_2 to be the product space

 Omega_2=product_(n in N)product_(z in Z^d)[0,infty)

and define associated to Omega_2 the usual product sigma-algebra and product measure P_2 where here, all the marginal probabilities are given by some probability measure mu on [0,infty). Finally, define Omega=Omega_1×Omega_2, equip Omega with the product measure P=P_1 square P_2 and usual product sigma-algebra. Under this construction, a Boolean model is a measurable mapping from Omega into N×Omega_2 defined by


where here, N denotes the set of all counting measures on the sigma-algebra B^d of Borel sets in R^d which assign finite measure to all bounded Borel sets and which assign values of at most 1 to points x in X.

One then transitions to percolation by first defining the collection of so-called binary cubes of order n


for all n in Z^+, z in Z^d, and by noting that each point x in X is contained in a unique binary cube K(n,z(n,x)) of order n. Moreover, for each x in X, there is a unique smallest number n_0=n_0(x) such that K(n_0,z(n_0,x)) contains no other points of X P_1-almost surely. This fact allows one to define the radius rho_x of the ball centered at x to be


where omega_2(n,z) is the notation used to denote an element omega_2 in Omega_2. Using this construction, one gets a collection of overlapping d-dimensional closed spheres whose radii are independent of the point process X and for which different points have balls with independent and identically-distributed radii.

It is not uncommon for a general Boolean model constructed in this way to be denoted (X,mu) or (X,rho), interchangeably. In the particular instance that X is a Poisson process with density lambda, the measure P is sometimes written P_lambda=P_((lambda,rho)) while the probability of an event A is then written P(A) or P{A} interchangeably.

In Boolean models, the space R^d is partitioned into two regions, namely the occupied region-the subset of R^d covered by at least one ball, denoted C-and its complement, the vacant region. These two regions are similar in that both consist of connected components (the occupied components and the vacant components, respectively) and the notation W(A) is used to denote the union of all occupied components having non-empty intersection with a subset A subset R^d. For A={0}, the notation W=W({0}) is used, and in the event of vacancy, the same notation is used throughout with V instead of W. Two points x,y in X which are in the same occupied component are said to be connected in the occupied region, sometimes denoted


Connectedness in the vacant region is defined analogously and denoted


If for some A subset R^d x and y are in the same occupied, respectively vacant, component of C intersection A, respectively of C^c intersection A, the notation x-->^oy in A, respectively x-->^vy in A is used.


The above figure illustrates a realization of a Boolean model, illustrating some of the terminology related to thereto. In this figure, the shaded region is C while the darker shaded region is W subset C. Note that V=V({0}) is empty due to the fact that W is non-empty. Moreover, the path joining x in C^c intersection A and y in C^c intersection A lies entirely in C^c intersection A; this indicates that x,y are in the same vacant component of C^c intersection A, whereby it follows that x-->^vy in A.

Historically, the term Boolean model was also used to refer to what's now known as the Boolean-Poisson model (Hanisch 1981).

See also

AB Percolation, Bernoulli Percolation Model, Boolean-Poisson Model, Bond Percolation, Bootstrap Percolation, Cayley Tree, Cluster, Cluster Perimeter, Continuum Percolation Theory, Dependent Percolation, Discrete Percolation Theory, Disk Model, First-Passage Percolation, Germ-Grain Model, Inhomogeneous Percolation Model, Lattice Animal, Long-Range Percolation Model, Mixed Percolation Model, Oriented Percolation Model, Percolation, Percolation Theory, Percolation Threshold, Polyomino, Random-Cluster Model, Random-Connection Model, Random Walk, s-Cluster, s-Run, Site Percolation

This entry contributed by Christopher Stover

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Hanisch, K. H. "On Classes of Random Sets and Point Process Models." Serdica Bulgariacae Mathematicae Publicationes 7, 160-166, 1981.Meester, R. and Roy, R. Continuum Percolation. New York: Cambridge University Press, 2008.

Cite this as:

Stover, Christopher. "Boolean Model." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein.

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