# On Limiting Gibbs States of the Two-Dimensional Ising Models

### Yasunari Higuchi

Kyoto University, Japan

## 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 $N_{+}/N_{–}<θ<3/5$ on every boundary is $μ_{–}$, where $N_{+}$ is the number of up-spins on the boundary and $N_{−}$ is that of down-spins. We also prove that for each $θ>3/5$, there exists a boundary condition such that $3/5<N_{+}/N_{–}≤θ$ on every boundary, and the limiting Gibbs state corresponding to this boundary condition is $μ_{+}$.

## Cite this article

Yasunari Higuchi, On Limiting Gibbs States of the Two-Dimensional Ising Models. Publ. Res. Inst. Math. Sci. 14 (1978), no. 1, pp. 53–69

DOI 10.2977/PRIMS/1195189280