Bernoulli Percolation Model

Intuitively, a model of d-dimensional percolation theory is said to be a Bernoulli model if the open/closed status of an area is completely random. In particular, it makes sense to talk about a Bernoulli bond percolation, Bernoulli site percolation, as well as describing other models of both discrete and continuum percolation theory as being Bernoulli.

Due to the vastness of the literature on percolation theory, however, there is a certain lack of uniformity in its terminology; as such, some authors choose to define d-dimensional Bernoulli percolation strictly in terms of its behavior on the standard bond percolation model within the regular point lattice Z^d. According to this view, the term Bernoulli percolation refers to the independent assignment as either open (with probability p in [0,1]) or closed (with probability 1-p) to each edge e in E^d where here,

 E^d={{x,y}:x,y in Z^d,|x-y|=1}.

This perspective, though framed relative to obvious graph theory terminology, is largely probabilistic (Cerf 2006).

See also

AB Percolation, Boolean Model, Boolean-Poisson Model, Bond Percolation, 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 Theory, Percolation Threshold, Polyomino, Random-Cluster Model, Random-Connection Model, Random Walk, s-Cluster, s-Run, Site Percolation

This entry contributed by Christopher Stover

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Cerf, R. In The Wulff Crystal in Ising and Percolation Models: Ecole d'Eté de Probabilités de Saint-Flour XXXIV-2004 (Ed. J. Picard). Netherlands: Springer-Verlag, 2006.Grimmett, G. Percolation, 2nd ed. Berlin: Springer-Verlag, 1999.

Cite this as:

Stover, Christopher. "Bernoulli Percolation Model." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein.

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