Dependent Percolation

The phrase dependent percolation is used in two-dimensional discrete percolation to describe any general model 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  is allowed to take on  different values rather than the usual two-and the random-cluster model.

Ostensibly, it would also make sense to talk about dependent percolation models in contexts such as -dimensional discrete percolation theory for arbitrary , as well as in continuum percolation theory; even so, the literature on such generalizations is scarce at best.

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." http://www.cts.cuni.cz/soubory/konference/pdf.pdf.Grimmett, G. Percolation, 2nd ed. Berlin: Springer-Verlag, 1999.

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 (Ed. H. W. Block, A. R. Sampson, and T. H. Savits). Hayward, CA: Institute of Mathematical Statistics, pp. 395-401, 1990.