Percolation and number of phases in the two-dimensional Ising model

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 stochastic monotonicity and avoids the use of symmetry 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 a homogeneous field and the hard-core lattice gas model on Z. © 2000 American Institute of Physics. @S0022-2488 ~00!00103-1#