Bernoulli Percolation Model
المؤلف:
Grimmett, G
المصدر:
Percolation, 2nd ed. Berlin: Springer-Verlag, 1999.
الجزء والصفحة:
...
13-5-2022
1818
Bernoulli Percolation Model
Intuitively, a model of 
-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 
-dimensional Bernoulli percolation strictly in terms of its behavior on the standard bond percolation model within the regular point lattice 
. According to this view, the term Bernoulli percolation refers to the independent assignment as either open (with probability ![p in [0,1]](https://mathworld.wolfram.com/images/equations/BernoulliPercolationModel/Inline4.svg)
) or closed (with probability 
) to each edge 
where here,
![E^d=<span style=]() {{x,y}:x,y in Z^d,|x-y|=1}. " src="https://mathworld.wolfram.com/images/equations/BernoulliPercolationModel/NumberedEquation1.svg" style="height:25px; width:247px" /> |
This perspective, though framed relative to obvious graph theory terminology, is largely probabilistic (Cerf 2006).
REFERENCES
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.
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