Translation invariant Gibbs states for the Ising model

We prove that all the translation invariant Gibbs states of the Ising model are a linear combination of the pure phases [InlineMediaObject not available: see fulltext.] for any [InlineMediaObject not available: see fulltext.]. This implies that the average magnetization is continuous for [InlineMediaObject not available: see fulltext.]. Furthermore, combined with previous results on the slab percolation threshold [B2] this shows the validity of Pisztora's coarse graining [Pi] up to the critical temperature.