Ising Model
المؤلف:
Balister, P. N.; Bollobás, B.; and Stacey, A. M
المصدر:
"Dependent Percolation in Two Dimensions." Prob. Theory Relat. Fields 117
الجزء والصفحة:
...
15-5-2022
1946
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
![X=cexp(sum_(<span style=]() {x,y})J_(xy)chi_A) " src="https://mathworld.wolfram.com/images/equations/IsingModel/NumberedEquation1.svg" style="height:57px; width:158px" /> |
where 
is a real constant, 
, and ![A=<span style=]()
{(x,y) in Z^2:T_x=T_y}" src="https://mathworld.wolfram.com/images/equations/IsingModel/Inline7.svg" style="height:25px; width:207px" /> for site random variables ![T_x,T_y in <span style=]()
{1,...,q}" src="https://mathworld.wolfram.com/images/equations/IsingModel/Inline8.svg" style="height:25px; width:140px" />, 
.
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.
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