Percolation and number of phases in the 2D Ising model

Details Percolation and number of phases in the 2D Ising model Percolation and number of phases in the 2D Ising model

1999-07-29

arxiv.org/abs/math/9907186v3

J. Math. Phys. 41, 1153-1169 (2000)

Mathematics

Probability

22 pages. Further details on extensions. To appear in J.Math.Phys., special issue on `Probabilistic Methods in Statistical Phy

Scientific paper

We reconsider the percolation approach of Russo, Aizenman and Higuchi for showing that there exist only two phases in the Ising model on the square lattice. We give a fairly short alternative proof which is only based on FKG monotonicity and avoids the use of GKS-type inequalities originally needed for some background results. Our proof extends to the Ising model on other planar lattices such as the triangular and honeycomb lattice. We can also treat the Ising antiferromagnet in an external field and the hard-core lattice gas model on $Z^2$.

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