Translation invariance and instability of phase coexistence in the two dimensional Ising system

It is shown that any Gibbs state of the two dimensional feromagnetic Ising system is of the form lambda μsub(+) + (1 - lambda)μsub(-), with some lambda element of [0,1]. This excludes the possibility of a locally stable phase coexistence and of translation symmetry breaking, which are known to occur in higher dimensions. Use is made in the proof of the stochastic aspects of the geometry of the interface lines. (orig.)