Ising Model

In statistical mechanics, the two-dimensional Ising model is a popular tool used to study the dipole moments of magnetic spins.

The Ising model in two dimensions is a type of dependent site percolation model which is characterized by the existence of a random variable  assigning to each point  a value of  and is driven by a distribution  of the form

where  is a real constant, , and  for site random variables , .

Some authors differentiate between positive (or ferromagnetic) dependency and negative (or antiferromagnetic) dependency (Newman 1990) depending on the sign of the quantity , though little mention of this distinction appears overall.

Other examples of dependent percolation models 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.

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.