In most modern literature, a Boolean model is a probabilistic model of continuum percolation theory characterized by the existence of a stationary
point process 
and a random variable 
which independently
determine the centers and the random radii
of a collection of closed balls in 
for some 
.
In this case, the model is said to be driven by 
.
Worth noting is that the most intuitive ideas about constructing a feasible model using 
and 
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 
and 
into a reasonable model of continuum percolation. The formal
construction is as follows.
Let 
be a stationary point process as discussed above and suppose that 
is defined on a probability
space 
.
Next, define the space 
to be the product space
 |
(1)
|
and define associated to 
the usual product sigma-algebra and product
measure 
where here, all the marginal probabilities
are given by some probability measure 
on 
. Finally, define 
, equip 
with the product measure 
and usual product 
-algebra. Under this construction, a Boolean model is a
measurable mapping from 
into 
defined by
 |
(2)
|
where here, 
denotes the set of all counting measures on the 
-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 defining the collection of so-called binary cubes of order 
![K(n,z)=product_(i=1)^d(z_i2^(-n),(z_i+1)2^(-n)]](/images/equations/BooleanModel/NumberedEquation3.svg) |
(3)
|
for all 
,
, and by noting that each point
is contained in a unique binary
cube 
of order 
.
Moreover, for each 
,
there is a unique smallest number 
such that 
contains no other points of 
-almost surely. This fact allows one to define the radius
of the ball centered at 
to be
 |
(4)
|
where 
is the notation used to denote an element 
. Using this construction, one gets a collection
of overlapping 
-dimensional
closed spheres whose radii are independent of the point
process 
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 
or 
, interchangeably. In the particular instance that 
is a Poisson
process with density 
,
the measure 
is sometimes written 
while the probability of an event 
is then written 
or 
interchangeably.
In Boolean models, the space 
is partitioned into two regions, namely the occupied region-the
subset of 
covered by at least one ball, denoted 
-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 
is used to denote the union of all occupied components
having non-empty intersection
with a subset 
. For 
, the notation 
is used, and in the event of vacancy, the same notation
is used throughout with 
instead of 
.
Two points 
which are in the same occupied component are said to be connected in the occupied
region, sometimes denoted
 |
(5)
|
Connectedness in the vacant region is defined analogously and denoted
 |
(6)
|
If for some 
and 
are in the same occupied, respectively vacant, component of
, respectively of 
, the notation 
in 
, respectively 
in 
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 
while the darker shaded region is 
. Note that 
is empty due to the fact that 
is non-empty. Moreover, the path joining 
and 
lies entirely in 
; this indicates that 
are in the same vacant component of 
, whereby it follows that 
in 
.
Historically, the term Boolean model was also used to refer to what's now known as the Boolean-Poisson model (Hanisch 1981).
