On the Absence of Non-Translationally Invariant Gibbs States in Two Dimensions

#### I. Merola

2000, v.6, Issue 4, 517-541

ABSTRACT

We consider an Ising system in $\nu=2$ dimensions with a ferromagnetic Kac potential whose scaling parameter is denoted by $\gamma$. We show that for any $\gamma$ sufficiently small every Gibbs state is translationally invariant. Taking account of the result on the phase diagram [P. Butta, I. Merola and E. Presutti, On the validity of the van der Waals theory, Markov Processes Relat. Fields, 1997, v.3, pp.63-88], our result implies that below the critical temperature, for any $\gamma$ sufficiently small there exist only two pure phases.

Keywords: ferromagnetic Kac potential,Ising system,translational invariance