Dependent percolation in discrete percolation theory consists of models in which the states of the various graph
edges (in the case of bond percolation models)
or graph vertices (in site
percolation models) are not independent .
Many models of this type come about naturally in a number of fields. For example, a popular tool in statistical mechanics is the two-dimensional Ising
model , a type of dependent site percolation
model used to study the dipole moments of magnetic spins. Other examples include
the Potts models -generalizations of the Ising
model in which random-cluster model .
Dependent percolation models also occur on graphs in arbitrary dimension and in continuum
percolation theory . Dependent random fields on countable graphs with bounded vertex degree
are studied by Liggett et al. (1997), and dependent continuum models built
from interacting germ-grain models are studied
by Coupier and Dereudre (2014).
See also AB Percolation,
Bernoulli Percolation Model ,
Bond Percolation ,
Boolean
Model ,
Boolean-Poisson Model ,
Bootstrap
Percolation ,
Cayley Tree ,
Cluster ,
Cluster Perimeter ,
Continuum
Percolation Theory ,
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 ,
Potts
Model ,
Random-Cluster Model ,
Random-Connection
Model ,
Random Walk ,
s -Cluster,
s -Run,
Site Percolation
Portions of this entry contributed by Christopher
Stover
Explore with Wolfram|Alpha
References Balister, P. N.; Bollobás, B.; and Stacey, A. M. "Dependent Percolation in Two Dimensions." Prob. Theory Relat. Fields 117 ,
495-513, 2000. Chayes, J. T.; Puha, A.; and Sweet, T. "Independent
and Dependent Percolation." https://www.cts.cuni.cz/soubory/konference/pdf.pdf . Coupier,
D. and Dereudre, D. "Continuum Percolation for Quermass Interaction Model."
Electron. J. Probab. 19 , Paper 35, 1-19, 2014. https://doi.org/10.1214/EJP.v19-2298 . Grimmett,
G. Percolation,
2nd ed. Liggett, T. M.;
Schonmann, R. H.; and Stacey, A. M. "Domination by Product Measures."
Ann. Probab. 25 , 71-95, 1997. https://doi.org/10.1214/aop/1024404279 . Newman,
C. M. "Ising Models and Dependent Percolation." In Topics
in Statistical Dependence. Proceedings of the Symposium on Dependence in Probability
and Statistics Held in Somerset, Pennsylvania, August 1-5, 1987 Referenced on Wolfram|Alpha Dependent Percolation
Cite this as:
Weisstein, Eric W. , with contributions by Christopher Stover . "Dependent Percolation." From MathWorld https://mathworld.wolfram.com/DependentPercolation.html
Subject classifications
is allowed to take on 
different values rather than the usual two-and the random-cluster model.
