Percolation theory deals with fluid flow (or any other similar process) in random media.
If the medium is a set of regular lattice points, then there are two main types of percolation: A site percolation considers the lattice vertices as the relevant entities; a bond percolation considers the lattice edges as the relevant entities. These two models are examples of discrete percolation theory, an umbrella term used to describe any percolation model which takes place on a regular point lattice or any other discrete set, and while they're most certainly the most-studied of the discrete models, others such as AB percolation and mixed percolation do exist and are reasonably well-studied in their own right.
Contrarily, one may also talk about continuum percolation models, i.e., models which attempt to define analogous tools and
results with respect to continuous, uncountable
domains. In particular, continuum percolation
theory involves notions of percolation for 
and for various non-discrete subsets
thereof. Unsurprisingly, there are a large number of models for continuum
percolation theory as well, most-studied among which are the Boolean,
Boolean-Poisson, disk,
and germ-grain models.
One of the most investigated aspects of percolation theory is the determination of a so-called percolation threshold; this problem is well-studied in both the discrete and continuum settings.
In the Season 2 episode "Soft Target" (2006) of the television crime drama NUMB3RS, character Charlie uses percolation theory to help locate the person who released potentially lethal gas into the Los Angeles subway system.
A dependent directed lattice model chooses a uniformly distributed 
-element subset of the 
neighbors independently at each vertex of 
, where 
, and places outgoing arrows to the chosen neighbors.
Liu et al. (2026) proved that the origin has an infinite outgoing cluster
with positive probability for every 
when 
. For 
that probability is zero. Thus the transition in the number
of chosen neighbors is exactly at two. This model is distinct from bootstrap
percolation, which evolves a set of occupied sites.
