The Boolean model for a random subset of the plane (or higher dimensions, analogously) is one of the simplest and most tractable models in stochastic geometry. Take a Poisson point process of rate in the plane and make each point be the center of a random set; the resulting union of overlapping sets is a realization of the Boolean model . More precisely, the parameters are and a probability distribution on compact sets; for each point of the Poisson point process we pick a set from the distribution, and then define as the union of translated sets.
To illustrate tractability with one simple formula, the mean density of equals where denotes the area of . The classical theory of stochastic geometry develops many further formulas – see [1] [2].
As related topics, the case of constant-sized discs is the basic model of continuum percolation [3] and the low-density Boolean models serve as a first-order approximations in the study of extremes in many models [4].