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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 a homogeneous field and the hard-core lattice gas model on Z 2 . Set-up and basic facts• the upper half-plane π up = {x = (x 1 , x 2 ) ∈ Z 2 : x 2 ≥ 0},

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Lem m a 2. 1. p o f .9 is a limiting Gibbsian measure if and only i f (2. 4) holds only f or any one point set V = { x } , x EZ ".