Translation invariant Gibbs states for the Ising model

Abstract

We prove that all the translation invariant Gibbs states of the Ising model are a linear combination of the pure phases ${\mu }^{+},{\mu }^{-}$ in the phase transition regime. This implies that the average magnetization is continuous in the phase transition regime ($\beta >{\beta }_c$). Furthermore, combined with previous results on the slab percolation threshold this shows the validity of Pisztora's coarse graining up to the critical temperature.

14 pages, 3 figures