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 sigma assigning to each point x=(x,y) in Z^2 a value of +/-1 and is driven by a distribution X=X(x,y) of the form


where c in R is a real constant, J_(xy) in R, and A={(x,y) in Z^2:T_x=T_y} for site random variables T_x,T_y in {1,...,q}, q in Z.

Some authors differentiate between positive (or ferromagnetic) dependency and negative (or antiferromagnetic) dependency (Newman 1990) depending on the sign of the quantity J_(xy), 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 sigma is allowed to take on n>=1 different values rather than the usual two-and the random-cluster model.

See also

Bond Percolation, Bootstrap Percolation, Continuum Percolation Theory, Percolation, Percolation Theory, Percolation Threshold, Site Percolation

This entry contributed by Christopher Stover

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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.", 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.

Cite this as:

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

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