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⁻<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⁺.