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.
Chayes, L. and Schonmann, R. H. "Mixed Percolation as a Bridge Between Site and Bond Percolation." Ann. Appl. Probab.10,
1182-1196, 2000.Deutscher, G.; Zallen, R.; and Adler, J. (Eds.). Percolation
Structures and Processes. Bristol: Adam Hilger, 1983.Grimmett,
G. Percolation.
New York: Springer-Verlag, 1989.Grimmett, G. Percolation and Disordered
Systems. Berlin: Springer-Verlag, 1997.Hammersley, J. M. "A
Generalization of McDiarmid's Theorem for Mixed Bernoulli Percolation." Math.
Proc. Camb. Phil. Soc.88, 167-170, 1980.Kesten, H. Percolation
Theory for Mathematicians. Boston, MA: Birkhäuser, 1982.Stauffer,
D. and Aharony, A. Introduction
to Percolation Theory, 2nd ed. London: Taylor & Francis, 1992.Weisstein,
E. W. "Books about Percolation Theory." http://www.ericweisstein.com/encyclopedias/books/PercolationTheory.html.
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.
