On Limiting Gibbs States of the Two-Dimensional Ising Models

Abstract

We consider limiting Gibbs states in the two-dimensional ferromagnetic Ising model at sufficiently low temperatures. We prove that every limiting Gibbs state corresponding to a boundary condition such that $N^{+}/N^{–}<\theta <3/5$ on every boundary is ${\mu }^{–}$, where $N^{+}$ is the number of up-spins on the boundary and $N^{-}$ is that of down-spins. We also prove that for each $\theta >3/5$, there exists a boundary condition such that $3/5<N^{+}/N^{–}\le \theta$ on every boundary, and the limiting Gibbs state corresponding to this boundary condition is ${\mu }^{+}$.