Continuum percolation theory studies percolation in Euclidean space 
and its nondiscrete subsets.
It can be thought of as a continuous, uncountable
version of percolation theory. In its most
studied discrete form, percolation theory takes
place on a countable point
lattice such as 
.
Models used to study continuum percolation theory include the disk model, the germ-grain model, and the random-connection
model. Perhaps the most well-studied is the Boolean-Poisson
model, which roughly consists of centering an independent
copy of a random 
-dimensional shape 
at each point of a homogeneous Poisson process 
in 
-dimensional Euclidean
space 
.
The resulting collection of overlapping shapes spans a subset
of 
.
Using this construction, one studies percolation
by considering whether a given random shape has positive probability of belonging to an infinite cluster of random shapes. Percolation
is defined similarly in the other models.
Gilbert (1961) introduced a continuum model to study communication networks of short-range stations spread over a wide area. Continuum percolation theory lacks much of the orderly mathematical structure of discrete percolation theory, whose methods based on enumeration lose much of their power in the continuum case. Even so, much work has been done to advance the field, which has applications in condensed matter and material physics (Hall 1985).
