Boolean model (probability theory)

Realization of Boolean model with random-radii discs.

In probability theory, 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 $\lambda$ 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 ${\mathcal {B}}$. More precisely, the parameters are $\lambda$ and a probability distribution on compact sets; for each point $\xi$ of the Poisson point process we pick a set $C_{\xi }$ from the distribution, and then define ${\mathcal {B}}$ as the union $\cup _{\xi }(\xi +C_{\xi })$ of translated sets.

To illustrate tractability with one simple formula, the mean density of ${\mathcal {B}}$ equals $1-\exp(-\lambda A)$ where $\Gamma$ denotes the area of $C_{\xi }$ and $A=\operatorname {E} (\Gamma ).$ The classical theory of stochastic geometry develops many further formulae.

As related topics, the case of constant-sized discs is the basic model of continuum percolation and the low-density Boolean models serve as a first-order approximations in the study of extremes in many models.