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# Percolation Theory

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.

AB Percolation, Bernoulli Percolation Model, Bond Percolation, Boolean Model, Boolean-Poisson Model, Bootstrap Percolation, Cayley Tree, Cluster, Cluster Perimeter, Continuum Percolation Theory, Dependent Percolation, Discrete Percolation Theory, Disk Model, First-Passage Percolation, Germ-Grain Model, Inhomogeneous Percolation Model, Lattice Animal, Long-Range Percolation Model, Mixed Percolation Model, Oriented Percolation Model, Percolation, Percolation Threshold, Polyomino, Random-Cluster Model, Random-Connection Model, Random Walk, s-Cluster, s-Run, Site Percolation

Portions of this entry contributed by Christopher Stover

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## References

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.

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Percolation Theory

## Cite this as:

Stover, Christopher and Weisstein, Eric W. "Percolation Theory." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/PercolationTheory.html